From 439f93c1a2cd59237b144aeb67deee87aaeccd03 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Fri, 26 May 2023 00:45:20 -0400 Subject: [PATCH 01/16] first pass at user guide --- docs/source/guide/pt-1-intro.md | 108 ++++++++++ docs/source/guide/pt-2-use-case.md | 32 +++ docs/source/guide/pt-3-options.md | 16 ++ docs/source/guide/pt-4-low-level.md | 92 ++++++++ docs/source/guide/pt-5-theory.md | 92 ++++++++ docs/source/guide/pt.md | 34 +++ docs/source/img/pt_workflow.svg | 313 ++++++++++++++++++++++++++++ mitiq/pt/__init__.py | 2 +- mitiq/pt/pt.py | 2 +- mitiq/pt/tests/test_pt.py | 6 +- 10 files changed, 692 insertions(+), 5 deletions(-) create mode 100644 docs/source/guide/pt-1-intro.md create mode 100644 docs/source/guide/pt-2-use-case.md create mode 100644 docs/source/guide/pt-3-options.md create mode 100644 docs/source/guide/pt-4-low-level.md create mode 100644 docs/source/guide/pt-5-theory.md create mode 100644 docs/source/guide/pt.md create mode 100644 docs/source/img/pt_workflow.svg diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md new file mode 100644 index 0000000000..a0f107d555 --- /dev/null +++ b/docs/source/guide/pt-1-intro.md @@ -0,0 +1,108 @@ +--- +jupytext: + text_representation: + extension: .myst + format_name: myst + format_version: 0.13 + jupytext_version: 1.11.1 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +--- + +# How do I use PT? + +As with all techniques, PT is compatible with any frontend supported by Mitiq: + +```{code-cell} ipython3 +import mitiq + +mitiq.SUPPORTED_PROGRAM_TYPES.keys() +``` + + +## Problem setup +We first define the circuit of interest. In this example, the circuit has a +slack window with a length of 4 (in the sense that 4 single-qubit gates can fit in that window). + +```{code-cell} ipython3 +from cirq import LineQubit, Circuit, rx, rz, CNOT + +a, b = LineQubit.range(2) +circuit = Circuit( + rx(0.1).on(a), + rx(0.1).on(a), + rz(0.4).on(a), + rx(-0.72).on(a), + rz(0.2).on(a), + rx(-0.8).on(b), + CNOT.on(a, b), +) + +print(circuit) +``` + +Next we define a simple executor function which inputs a circuit, executes +the circuit on a noisy simulator, and returns the probability of the ground +state. See the [Executors](executors.md) section for more information on +how to define more advanced executors. + +```{code-cell} ipython3 +import numpy as np +from cirq import DensityMatrixSimulator, amplitude_damp +from mitiq.interface import convert_to_mitiq + +def execute(circuit, noise_level=0.1): + """Returns Tr[ρ |0⟩⟨0|] where ρ is the state prepared by the circuit + executed with amplitude damping noise. + """ + # Replace with code based on your frontend and backend. + mitiq_circuit, _ = convert_to_mitiq(circuit) + noisy_circuit = mitiq_circuit.with_noise(amplitude_damp(gamma=noise_level)) + rho = DensityMatrixSimulator().simulate(noisy_circuit).final_density_matrix + return rho[0, 0].real +``` + +The [executor](executors.md) can be used to evaluate noisy (unmitigated) +expectation values. + +```{code-cell} ipython3 +# Compute the expectation value of the |0><0| observable. +noisy_value = execute(circuit) +ideal_value = execute(circuit, noise_level=0.0) +print(f"Error without mitigation: {abs(ideal_value - noisy_value) :.3}") +``` + +## Apply PT +Pauli Twirling can be easily implemented with the function +{func}`.execute_with_pt()`. + +```{code-cell} ipython3 +mitigated_result = pt.execute_with_pt( + circuit=circuit, + executor=execute, + rule=rule, +) +``` + +```{code-cell} ipython3 +print(f"Error with mitigation (PT): {abs(ideal_value - mitigated_result) :.3}") +``` + +Here we observe that the application of PT reduces the estimation error when compared +to the unmitigated result. + +```{admonition} Note: +PT is designed to mitigate noise that has a finite correlation time. For the +simple Markovian noise simulated in this example, PT can still have a +non-trivial effect on the final error, but it is not always a positive effect. +For example, one can check that by changing the parameters of the input circuit, +the error with PT is sometimes larger than the unmitigated error. +``` + ++++ + +The section +[What additional options are available when using PT?](pt-3-options.md) +contains information on more advanced ways of applying PT with Mitiq. diff --git a/docs/source/guide/pt-2-use-case.md b/docs/source/guide/pt-2-use-case.md new file mode 100644 index 0000000000..5a1ddd3d72 --- /dev/null +++ b/docs/source/guide/pt-2-use-case.md @@ -0,0 +1,32 @@ +--- +jupytext: + text_representation: + extension: .myst + format_name: myst + format_version: 0.13 + jupytext_version: 1.10.3 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +--- + +# When should I use PT? + +## Advantages + +Pauli Twirling is a technique devised to minimize Markovian noise, by tailoring it towards Pauli channels. + +More details on the theory of Pauli Twirling are given in the section [What is the theory behind PT?](pt-5-theory.md). + +Pauli Twirling is agnostic to our knowledge on the type of noise, easy to implement, and useful to better understand and minimize the benchmarking vs performance gap. + + + +## Disadvantages + +Pauli Twirling is generally combined with a recompilation to maintain circuit depth. Mitiq thus far does not provide this compilation, and so circuit depth is increased by the additional single-qubit gates. + +Though the noise is tailored towards a more mitigable channel, it's possible for this channel to be entirely noisy (i.e. a completely depolarizing channel). + +Furthermore, though there are extensions to account for memory effects, the basic implementation in Mitiq should not be expected to mitigate non-Markovian noise (on its own). diff --git a/docs/source/guide/pt-3-options.md b/docs/source/guide/pt-3-options.md new file mode 100644 index 0000000000..e0b944dede --- /dev/null +++ b/docs/source/guide/pt-3-options.md @@ -0,0 +1,16 @@ +--- +jupytext: + text_representation: + extension: .myst + format_name: myst + format_version: 0.13 + jupytext_version: 1.11.1 +kernelspec: + display_name: Python 3 (ipykernel) + language: python + name: python3 +--- + +# What additional options are available when using PT? + +Currently Pauli Twirling is designed to have relatively few options, in part to make it readily composable with every other Mitiq technique. In the future, we expect a possibility of supporting additional operations as targets (beyond CZ and CNOT gates), with more customization on picking those targets. Stay tuned! \ No newline at end of file diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md new file mode 100644 index 0000000000..7d1060927f --- /dev/null +++ b/docs/source/guide/pt-4-low-level.md @@ -0,0 +1,92 @@ +--- +jupytext: + text_representation: + extension: .myst + format_name: myst + format_version: 0.13 + jupytext_version: 1.11.1 +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +# What happens when I use PT? + +The workflow of Pauli Twirling (PT) in Mitiq is represented in the figure below. + +```{figure} ../img/pt_workflow.svg +--- +width: 700px +name: pt-workflow-overview +--- +Workflow of the PT technique in Mitiq, detailed in the [What happens when I use PT?](pt-4-low-level.md) section. +``` + +- The user provides a `QPROGRAM`, (i.e. a quantum circuit defined via any of the supported [frontends](frontends-backends.md)). +- Mitiq modifies the input circuit with the insertion of PT gates on noisy operations. +- The modified circuit is executed via a user-defined [Executor](executors.md). +- The error mitigated expectation value is returned to the user. + +With respect to the workflows of other error-mitigation techniques (e.g. [ZNE](zne-4-low-level.md) or [PEC](pec-4-low-level.md)), +PT involves the generation and the execution of a _single_ modified circuit. +For this reason, there is no need to combine the results of multiple circuits and the final inference step which is necessary for other +techniques is instead trivial for PT. + +```{note} +When setting the `num_trials` option to a value larger than one, multiple circuits are actually generated by Mitiq and +the associated results are averaged to obtain the final expectation value. This more general case is not shown in the figure since +it can be considered as an average of independent single-circuit workflows. +``` + +As shown in [How do I use PT?](pt-1-intro.md), the function {func}`.execute_with_pt()` applies PT behind the scenes +and directly returns the error-mitigated expectation value. +In the next sections instead, we show how one can apply PT at a lower level, i.e., by: + +- Detecting CZ and CNOT gates in the circuit; +- Mapping those operations to their twirled equivalences; +- Executing the modified circuit. + + +## Executing the modified circuit + +```{code-cell} ipython3 +from cirq import DensityMatrixSimulator, amplitude_damp + +def execute(circuit, noise_level=0.003): + """Returns Tr[ρ |00..⟩⟨00..|] where ρ is the state prepared by the circuit + executed with depolarizing noise. + """ + noisy_circuit = circuit.with_noise(amplitude_damp(noise_level)) + rho = DensityMatrixSimulator().simulate(noisy_circuit).final_density_matrix + return rho[0, 0].real +``` + +If executed on a noiseless backend, `circuit_with_pt` and `circuit` are equivalent. +On a real backend, they have a different sensitivity to noise. The core idea of the PT technique is that, +`circuit_with_pt` is (hopefully) less sensitive Markovian to noise thanks to the randomized Pauli insertions. + +```{code-cell} ipython3 +# Ideal result +execute(circuit, noise_level=0) +``` + +```{code-cell} ipython3 +# Unmitigated result +execute(circuit) +``` + +```{code-cell} ipython3 +# Error-mitigated result (with PT) +execute(circuit_with_pt) +``` + +As a final remark, we stress that the low-level procedure that we have shown is exactly what {func}`.execute_with_ddd()` does behind the scenes. +Let's verify this fact: + +```{code-cell} ipython3 +np.isclose( + pt.execute_with_pt(circuit, execute, rule=xyxy_rule), + execute(circuit_with_pt), +) +``` diff --git a/docs/source/guide/pt-5-theory.md b/docs/source/guide/pt-5-theory.md new file mode 100644 index 0000000000..1eb55781b5 --- /dev/null +++ b/docs/source/guide/pt-5-theory.md @@ -0,0 +1,92 @@ +--- +jupytext: + text_representation: + extension: .myst + format_name: myst + format_version: 0.13 + jupytext_version: 1.11.1 +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +# What is the theory behind DDD? + +Dynamical decoupling (DD) {cite}`Viola_1998_PRA, Viola_1999_PRL, Zhang_2014_PRL` +is a quantum control technique to effectively reduce the interaction of a quantum system with its environment. +The protocol works by driving a quantum system with rapid sequences of periodic control pulses. + +The application of DD sequences can have two effects depending on the correlation time {cite}`Breuer_2007_Oxford` of the environment: + +1. For Markovian noise, DD can make the overall quantum channel more symmetric (analogous to quantum twirling {cite}`Wallman_2016_PRA`) +but cannot actually decouple the system from the environment; + +2. For non-Markovian noise, DD can effectively decouple the system from the environment. +In theory, ideal sequences of infinitely quick and strong pulses, can result in complete noise suppression. + +In practice, due to the finite frequency and finite amplitude of DD sequences, +both effects are possible but only as imperfect approximations. + +In the context of quantum computing, DD can be considered as an error mitigation method. +With respect to other error mitigation techniques, DD has very peculiar features: + +- It maps a noisy quantum computation to a _single_ error-mitigated computation (no need to take linear combinations +of noisy results as in [ZNE](zne-5-theory.md [PEC](pec-5-theory.md) and [CDR](cdr-5-theory.md)). + +- As a consequence of the previous point, there is not a fundamental error mitigation overhead or +increase in statistical uncertainty in the final result. + +- If noise is time-correlated, it can suppress real errors at the physical level instead of applying a virtual noise +reduction via classical post-processing. + + + + +## Digital dynamical decoupling + +In a quantum computing device based on the circuit model, sequences of DD pulses can be mapped to sequences +of discrete quantum gates (typically Pauli gates). We refer to this gate-level formulation as _digital dynamical decoupling_ (DDD) +to distinguish it from the standard pulse-level formulation. + + + +```{note} +This type of gate-level approach is very similar to the gate-level abstraction used in Mitiq to implement +_digital zero-noise extrapolation_ via _unitary folding_ (see [What is the theory behind ZNE?](zne-5-theory.md)). +``` +Experimental evidence showing the practical utility gate-level decoupling sequences is given in several publications {cite}`Pokharel_2018_PRL, Jurcevic_2021_arxiv, GoogleQuantum_2021_nature, Smith_2021_arxiv, Das_2021_ACM`. + + +```{warning} +Gate-level DDD can only be considered as an approximation of the ideal (pulse-level) DD technique. Moreover, quantum backends +may internally optimize and schedule gates in unpredictable ways such that, in practice, DDD sequences may not be physically applied +as expected. +``` + +A significant advantage of DDD with respect to pulse-level DD is the possibility of defining it in a backend-independent way, +via simple transformations of abstract quantum circuits. For this reason, DDD is particularly suitable for a multi-platform library like Mitiq. + + + +## Common examples of DDD sequences + +Common dynamical decoupling sequences are arrays of (evenly spaced) Pauli gates. In particular: +- The _XX_ sequence is typically appropriate for mitigating (time-correlated) dephasing noise; +- The _YY_ sequence is typically appropriate for mitigating (time-correlated) amplitude damping noise; +- The _XYXY_ sequence is typically appropriate for mitigating generic single-qubit noise. + +```{note} +A general property of DDD sequences is that, if executed on a noiseless backend, they are equivalent to the identity operation. +``` + +All the above examples of DDD sequences are supported in Mitiq and more general ones can be defined and customized by users. +For more details on how to define DDD sequences in Mitiq see [What additional options are available for DDD?](ddd-3-options.md). + +In a practical scenario it is hard to characterize the noise model and the noise spectrum of a quantum device and the +choice of the optimal sequence is not obvious _a priori_. A possible strategy is to run a few circuits whose noiseless +results are theoretically known, such that one can empirically determine what sequence is optimal for a specific backend. + +It may happen that, for some sequences, the final error of the quantum computation is actually increased. +As with all other error-mitigation techniques, one should always take into account that an improvement of performances is not guaranteed. + diff --git a/docs/source/guide/pt.md b/docs/source/guide/pt.md new file mode 100644 index 0000000000..20cf78858c --- /dev/null +++ b/docs/source/guide/pt.md @@ -0,0 +1,34 @@ +# Pauli Twirling + +Pauli Twirling (PT) is an error mitigation technique in which, +in the Mitiq implementation, particularly noisy operations (e.g. CZ and CNOT) +are mitigated by independent, random, single-qubit gates inserted into +the circuit such that the effective logical circuit remains unchanged +but the noise is tailored into stochastic Pauli errors, as opposed to +the Markovian processes that are otherwise dominant. +For more discussion of the theory of PT, see the section [What is the theory +behind PT?](pt-5-theory.md). + +```{figure} ../img/pt_workflow.svg +--- +width: 700px +name: pt-workflow-overview +--- +Workflow of the PT technique in Mitiq, detailed in the [What happens when I use PT?](pt-4-low-level.md) section. +``` + +Below you can find sections of the documentation that address the following questions: + +```{toctree} +--- +maxdepth: 1 +--- +pt-1-intro.md +pt-2-use-case.md +pt-3-options.md +pt-4-low-level.md +pt-5-theory.md +``` + +You can find many more examples on a variety of error mitigation techniques in the **[Examples](../examples/examples.md)** section of +the documentation. diff --git a/docs/source/img/pt_workflow.svg b/docs/source/img/pt_workflow.svg new file mode 100644 index 0000000000..7d1484685c --- /dev/null +++ b/docs/source/img/pt_workflow.svg @@ -0,0 +1,313 @@ + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + User + + mitiq.ddd + + + + + Hardware + + + + + + + 2. Inference (trivial for PT) + + + + 1. Generate circuit(s) + + Circuit withtwirling gatesaround noisy gates + Error-mitigatedexpectationvalue + Execute onbackend + Quantumprogram + + diff --git a/mitiq/pt/__init__.py b/mitiq/pt/__init__.py index 046ce9bcae..7d73a7bd1e 100644 --- a/mitiq/pt/__init__.py +++ b/mitiq/pt/__init__.py @@ -4,7 +4,7 @@ # LICENSE file in the root directory of this source tree. from mitiq.pt.pt import ( - execute_with_pauli_twirling, + execute_with_pt, twirl_CNOT_gates, twirl_CZ_gates, ) diff --git a/mitiq/pt/pt.py b/mitiq/pt/pt.py index cf9af62cb2..d12b445bf3 100644 --- a/mitiq/pt/pt.py +++ b/mitiq/pt/pt.py @@ -51,7 +51,7 @@ ] -def execute_with_pauli_twirling( +def execute_with_pt( circuit: QPROGRAM, executor: Union[Executor, Callable[[QPROGRAM], QuantumResult]], observable: Optional[Observable] = None, diff --git a/mitiq/pt/tests/test_pt.py b/mitiq/pt/tests/test_pt.py index a06f5ba693..4e3f1a4be6 100644 --- a/mitiq/pt/tests/test_pt.py +++ b/mitiq/pt/tests/test_pt.py @@ -13,7 +13,7 @@ from mitiq.pt.pt import ( twirl_CNOT_gates, twirl_CZ_gates, - execute_with_pauli_twirling, + execute_with_pt, CNOT_twirling_gates, CZ_twirling_gates, ) @@ -116,7 +116,7 @@ def test_twirl_CNOT_increases_layer_count(): assert num_gates_after == num_gates_before -def test_execute_with_pauli_twirling(): +def test_execute_with_pt(): num_qubits = 3 num_layers = 20 circuit, _ = generate_mirror_circuit( @@ -124,7 +124,7 @@ def test_execute_with_pauli_twirling(): two_qubit_gate_prob=1.0, connectivity_graph=nx.complete_graph(num_qubits), ) - expval = execute_with_pauli_twirling( + expval = execute_with_pt( circuit, amp_damp_executor, num_circuits=10 ) assert 0 <= expval < 0.4 From adb6919b7f5e4b2985695c154ca9c815895d6f96 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Fri, 26 May 2023 12:12:52 -0400 Subject: [PATCH 02/16] format fix --- mitiq/pt/tests/test_pt.py | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/mitiq/pt/tests/test_pt.py b/mitiq/pt/tests/test_pt.py index 4e3f1a4be6..a6c728e5d0 100644 --- a/mitiq/pt/tests/test_pt.py +++ b/mitiq/pt/tests/test_pt.py @@ -124,7 +124,5 @@ def test_execute_with_pt(): two_qubit_gate_prob=1.0, connectivity_graph=nx.complete_graph(num_qubits), ) - expval = execute_with_pt( - circuit, amp_damp_executor, num_circuits=10 - ) + expval = execute_with_pt(circuit, amp_damp_executor, num_circuits=10) assert 0 <= expval < 0.4 From 2e22af10460d35084eab87f8c7ade35e770e686b Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Mon, 29 May 2023 21:10:05 -0400 Subject: [PATCH 03/16] wrap up on pt user guide --- docs/source/guide/guide.md | 1 + docs/source/guide/pt-1-intro.md | 20 +++----- docs/source/guide/pt-2-use-case.md | 2 +- docs/source/guide/pt-4-low-level.md | 6 +-- docs/source/guide/pt-5-theory.md | 79 ++++++----------------------- docs/source/refs.bib | 30 +++++++++++ 6 files changed, 58 insertions(+), 80 deletions(-) diff --git a/docs/source/guide/guide.md b/docs/source/guide/guide.md index a0fd1f0947..8568cd590f 100644 --- a/docs/source/guide/guide.md +++ b/docs/source/guide/guide.md @@ -10,6 +10,7 @@ pec.md cdr.md ddd.md rem.md +pt.md error-mitigation.md glossary.md ``` diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md index a0f107d555..24a64af5b2 100644 --- a/docs/source/guide/pt-1-intro.md +++ b/docs/source/guide/pt-1-intro.md @@ -29,15 +29,11 @@ slack window with a length of 4 (in the sense that 4 single-qubit gates can fit ```{code-cell} ipython3 from cirq import LineQubit, Circuit, rx, rz, CNOT -a, b = LineQubit.range(2) +a, b, c, d = LineQubit.range(4) circuit = Circuit( - rx(0.1).on(a), - rx(0.1).on(a), - rz(0.4).on(a), - rx(-0.72).on(a), - rz(0.2).on(a), - rx(-0.8).on(b), CNOT.on(a, b), + CZ.on(b, c), + CNOT.on(c, d), ) print(circuit) @@ -94,11 +90,11 @@ Here we observe that the application of PT reduces the estimation error when com to the unmitigated result. ```{admonition} Note: -PT is designed to mitigate noise that has a finite correlation time. For the -simple Markovian noise simulated in this example, PT can still have a -non-trivial effect on the final error, but it is not always a positive effect. -For example, one can check that by changing the parameters of the input circuit, -the error with PT is sometimes larger than the unmitigated error. +PT is designed to mitigate the simple Markovian noise simulated in this example, +but PT can still have a non-trivial effect on the final error, but it is not +always a positive effect. For example, one can check that by changing the +parameters of the input circuit, the error with PT is sometimes larger than +the unmitigated error. ``` +++ diff --git a/docs/source/guide/pt-2-use-case.md b/docs/source/guide/pt-2-use-case.md index 5a1ddd3d72..707ac07961 100644 --- a/docs/source/guide/pt-2-use-case.md +++ b/docs/source/guide/pt-2-use-case.md @@ -25,7 +25,7 @@ Pauli Twirling is agnostic to our knowledge on the type of noise, easy to implem ## Disadvantages -Pauli Twirling is generally combined with a recompilation to maintain circuit depth. Mitiq thus far does not provide this compilation, and so circuit depth is increased by the additional single-qubit gates. +Pauli Twirling is generally combined with a compilation pass to maintain circuit depth. Mitiq thus far does not provide this compilation, and so circuit depth is increased by the additional single-qubit gates. Though the noise is tailored towards a more mitigable channel, it's possible for this channel to be entirely noisy (i.e. a completely depolarizing channel). diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index 7d1060927f..c0259b9887 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -64,7 +64,7 @@ def execute(circuit, noise_level=0.003): If executed on a noiseless backend, `circuit_with_pt` and `circuit` are equivalent. On a real backend, they have a different sensitivity to noise. The core idea of the PT technique is that, -`circuit_with_pt` is (hopefully) less sensitive Markovian to noise thanks to the randomized Pauli insertions. +`circuit_with_pt` is (hopefully) less sensitive to Markovian noise thanks to the randomized Pauli insertions. ```{code-cell} ipython3 # Ideal result @@ -81,12 +81,12 @@ execute(circuit) execute(circuit_with_pt) ``` -As a final remark, we stress that the low-level procedure that we have shown is exactly what {func}`.execute_with_ddd()` does behind the scenes. +As a final remark, we stress that the low-level procedure that we have shown is exactly what {func}`.execute_with_pt()` does behind the scenes. Let's verify this fact: ```{code-cell} ipython3 np.isclose( - pt.execute_with_pt(circuit, execute, rule=xyxy_rule), + pt.execute_with_pt(circuit, execute), execute(circuit_with_pt), ) ``` diff --git a/docs/source/guide/pt-5-theory.md b/docs/source/guide/pt-5-theory.md index 1eb55781b5..1534c936d3 100644 --- a/docs/source/guide/pt-5-theory.md +++ b/docs/source/guide/pt-5-theory.md @@ -11,25 +11,27 @@ kernelspec: name: python3 --- -# What is the theory behind DDD? +# What is the theory behind Pauli Twirling? -Dynamical decoupling (DD) {cite}`Viola_1998_PRA, Viola_1999_PRL, Zhang_2014_PRL` -is a quantum control technique to effectively reduce the interaction of a quantum system with its environment. -The protocol works by driving a quantum system with rapid sequences of periodic control pulses. +Pauli Twirling (PT) {cite}`Wallman_2016_PRA Hashim_2021_PRX Urbanek_2021_PRL` +is a quantum error mitigation technique designed to tailor noise from a Markovian +channel to a more manageable stochastic Pauli channel. This tailoring is achieved +by randomly applying a series of Pauli operations to the quantum system, and in doing +so reduces the complexity of the noise channel. The application of DD sequences can have two effects depending on the correlation time {cite}`Breuer_2007_Oxford` of the environment: -1. For Markovian noise, DD can make the overall quantum channel more symmetric (analogous to quantum twirling {cite}`Wallman_2016_PRA`) -but cannot actually decouple the system from the environment; +1. For Markovian noise, PT can make the overall quantum channel more symmetric (analogous to dynamical decoupling {cite}`Viola_1998_PRA, Viola_1999_PRL, Zhang_2014_PRL`) -2. For non-Markovian noise, DD can effectively decouple the system from the environment. -In theory, ideal sequences of infinitely quick and strong pulses, can result in complete noise suppression. +2. For non-Markovian noise, PT's performance might be degraded due to the inability to completely decouple the system from the environment in the presence of memory effects -In practice, due to the finite frequency and finite amplitude of DD sequences, -both effects are possible but only as imperfect approximations. +Pauli Twirling (PT) can be a powerful tool for noise management in quantum systems. By twirling over the Pauli gates, PT transforms complex noise channels into simpler stochastic Pauli noise channels. Yet, as with all powerful tools, careful handling is required. -In the context of quantum computing, DD can be considered as an error mitigation method. -With respect to other error mitigation techniques, DD has very peculiar features: +The success of PT is contingent on various factors, such as the nature of the noise and the specific characteristics of the quantum system. It's worth noting that, while PT generally simplifies the noise channel, there are circumstances where it could transform the noise negatively, for example into a completely depolarizing channel with a corresponding total loss of quantum information. + +For optimal results, Pauli Twirling should be implemented with an understanding of the underlying noise dynamics, and ideally, should be complemented with other error mitigation techniques to ensure robust quantum computation. + +In the context of quantum error mitigation, PT and DDD stand apart in that they have very peculiar features with respect to other error mitigation techniques. PT's peculiarities include: - It maps a noisy quantum computation to a _single_ error-mitigated computation (no need to take linear combinations of noisy results as in [ZNE](zne-5-theory.md [PEC](pec-5-theory.md) and [CDR](cdr-5-theory.md)). @@ -37,56 +39,5 @@ of noisy results as in [ZNE](zne-5-theory.md [PEC](pec-5-theory.md) and [CDR](cd - As a consequence of the previous point, there is not a fundamental error mitigation overhead or increase in statistical uncertainty in the final result. -- If noise is time-correlated, it can suppress real errors at the physical level instead of applying a virtual noise -reduction via classical post-processing. - - - - -## Digital dynamical decoupling - -In a quantum computing device based on the circuit model, sequences of DD pulses can be mapped to sequences -of discrete quantum gates (typically Pauli gates). We refer to this gate-level formulation as _digital dynamical decoupling_ (DDD) -to distinguish it from the standard pulse-level formulation. - - - -```{note} -This type of gate-level approach is very similar to the gate-level abstraction used in Mitiq to implement -_digital zero-noise extrapolation_ via _unitary folding_ (see [What is the theory behind ZNE?](zne-5-theory.md)). -``` -Experimental evidence showing the practical utility gate-level decoupling sequences is given in several publications {cite}`Pokharel_2018_PRL, Jurcevic_2021_arxiv, GoogleQuantum_2021_nature, Smith_2021_arxiv, Das_2021_ACM`. - - -```{warning} -Gate-level DDD can only be considered as an approximation of the ideal (pulse-level) DD technique. Moreover, quantum backends -may internally optimize and schedule gates in unpredictable ways such that, in practice, DDD sequences may not be physically applied -as expected. -``` - -A significant advantage of DDD with respect to pulse-level DD is the possibility of defining it in a backend-independent way, -via simple transformations of abstract quantum circuits. For this reason, DDD is particularly suitable for a multi-platform library like Mitiq. - - - -## Common examples of DDD sequences - -Common dynamical decoupling sequences are arrays of (evenly spaced) Pauli gates. In particular: -- The _XX_ sequence is typically appropriate for mitigating (time-correlated) dephasing noise; -- The _YY_ sequence is typically appropriate for mitigating (time-correlated) amplitude damping noise; -- The _XYXY_ sequence is typically appropriate for mitigating generic single-qubit noise. - -```{note} -A general property of DDD sequences is that, if executed on a noiseless backend, they are equivalent to the identity operation. -``` - -All the above examples of DDD sequences are supported in Mitiq and more general ones can be defined and customized by users. -For more details on how to define DDD sequences in Mitiq see [What additional options are available for DDD?](ddd-3-options.md). - -In a practical scenario it is hard to characterize the noise model and the noise spectrum of a quantum device and the -choice of the optimal sequence is not obvious _a priori_. A possible strategy is to run a few circuits whose noiseless -results are theoretically known, such that one can empirically determine what sequence is optimal for a specific backend. - -It may happen that, for some sequences, the final error of the quantum computation is actually increased. -As with all other error-mitigation techniques, one should always take into account that an improvement of performances is not guaranteed. +- If noise is time-correlated, PT should not be expected to yield positive (noise-reducing) results on its own diff --git a/docs/source/refs.bib b/docs/source/refs.bib index 36cdc2945e..fbfc70cbc9 100644 --- a/docs/source/refs.bib +++ b/docs/source/refs.bib @@ -284,6 +284,21 @@ @article{GoogleQuantum_2021_nature } # Letter H +@article{Hashim_2021_PRX, + title = {Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor}, + author = {Hashim, Akel and Naik, Ravi K. and Morvan, Alexis and Ville, Jean-Loup and Mitchell, Bradley and Kreikebaum, John Mark and Davis, Marc and Smith, Ethan and Iancu, Costin and O'Brien, Kevin P. and Hincks, Ian and Wallman, Joel J. and Emerson, Joseph and Siddiqi, Irfan}, + journal = {Phys. Rev. X}, + volume = {11}, + issue = {4}, + pages = {041039}, + numpages = {12}, + year = {2021}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevX.11.041039}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.11.041039} +} + # Letter I # Letter J @@ -708,6 +723,21 @@ @article{Temme_2017_PRL # Letter U +@article{Urbanek_2021_PRL, + title = {Mitigating Depolarizing Noise on Quantum Computers with Noise-Estimation Circuits}, + author = {Urbanek, Miroslav and Nachman, Benjamin and Pascuzzi, Vincent R. and He, Andre and Bauer, Christian W. and de Jong, Wibe A.}, + journal = {Phys. Rev. Lett.}, + volume = {127}, + issue = {27}, + pages = {270502}, + numpages = {6}, + year = {2021}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.127.270502}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.127.270502} +} + # Letter V @article{Viola_1999_PRL, title = {Dynamical Decoupling of Open Quantum Systems}, From 89daa56d1b80c893420098074a7637ec46b28ef5 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Mon, 29 May 2023 22:41:10 -0400 Subject: [PATCH 04/16] missed save --- docs/source/guide/pt-5-theory.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/source/guide/pt-5-theory.md b/docs/source/guide/pt-5-theory.md index 1534c936d3..dfc2599bcc 100644 --- a/docs/source/guide/pt-5-theory.md +++ b/docs/source/guide/pt-5-theory.md @@ -13,7 +13,7 @@ kernelspec: # What is the theory behind Pauli Twirling? -Pauli Twirling (PT) {cite}`Wallman_2016_PRA Hashim_2021_PRX Urbanek_2021_PRL` +Pauli Twirling (PT) {cite}`Wallman_2016_PRA, Hashim_2021_PRX, Urbanek_2021_PRL` is a quantum error mitigation technique designed to tailor noise from a Markovian channel to a more manageable stochastic Pauli channel. This tailoring is achieved by randomly applying a series of Pauli operations to the quantum system, and in doing From f2ec63d71c1e20a225e95b739efeb1fba6f433fd Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Mon, 29 May 2023 22:46:35 -0400 Subject: [PATCH 05/16] missed docs iomport --- docs/source/guide/pt-1-intro.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md index 24a64af5b2..8b07a74851 100644 --- a/docs/source/guide/pt-1-intro.md +++ b/docs/source/guide/pt-1-intro.md @@ -27,7 +27,7 @@ We first define the circuit of interest. In this example, the circuit has a slack window with a length of 4 (in the sense that 4 single-qubit gates can fit in that window). ```{code-cell} ipython3 -from cirq import LineQubit, Circuit, rx, rz, CNOT +from cirq import LineQubit, Circuit, CZ, CNOT a, b, c, d = LineQubit.range(4) circuit = Circuit( From 9493c02e3fd00414f686910d0a211215c2d012b1 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Mon, 29 May 2023 23:53:24 -0400 Subject: [PATCH 06/16] missed docs import again --- docs/source/guide/pt-1-intro.md | 1 + 1 file changed, 1 insertion(+) diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md index 8b07a74851..705bdd6c75 100644 --- a/docs/source/guide/pt-1-intro.md +++ b/docs/source/guide/pt-1-intro.md @@ -75,6 +75,7 @@ Pauli Twirling can be easily implemented with the function {func}`.execute_with_pt()`. ```{code-cell} ipython3 +from mitiq import pt mitigated_result = pt.execute_with_pt( circuit=circuit, executor=execute, From 273e4539629bf92b52ac171d8b3374f446c610c5 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 05:47:13 -0400 Subject: [PATCH 07/16] missed erroneous var --- docs/source/guide/pt-1-intro.md | 1 - 1 file changed, 1 deletion(-) diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md index 705bdd6c75..c1990b654c 100644 --- a/docs/source/guide/pt-1-intro.md +++ b/docs/source/guide/pt-1-intro.md @@ -79,7 +79,6 @@ from mitiq import pt mitigated_result = pt.execute_with_pt( circuit=circuit, executor=execute, - rule=rule, ) ``` From e5fd05d87187ac91ea22fb5cd2e0a6c1db0ccf51 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 08:49:18 -0400 Subject: [PATCH 08/16] clarify fourth section and cleanup errors --- docs/source/guide/pt-4-low-level.md | 31 +++++++++++++++++++++++++++-- 1 file changed, 29 insertions(+), 2 deletions(-) diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index c0259b9887..654cb92d3d 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -43,10 +43,37 @@ As shown in [How do I use PT?](pt-1-intro.md), the function {func}`.execute_with and directly returns the error-mitigated expectation value. In the next sections instead, we show how one can apply PT at a lower level, i.e., by: -- Detecting CZ and CNOT gates in the circuit; -- Mapping those operations to their twirled equivalences; +- Twirling CZ and CNOT gates in the circuit; - Executing the modified circuit. +## Twirling CZ and CNOT gates in the circuit +First let's define our circuit: +```{code-cell} ipython3 +from cirq import LineQubit, Circuit, CZ, CNOT +fr + +a, b, c, d = LineQubit.range(4) +circuit = Circuit( + CNOT.on(a, b), + CZ.on(b, c), + CNOT.on(c, d), +) + +print(circuit) +``` +Now, we can see what happens when we apply the PT functions, through {func}`.twirl_CNOT_gates()` and the subsequent {func}`.twirl_CZ_gates()` +```{code-cell} ipython3 +from mitiq import pt + +circuit_to_twirl = circuit.copy() +CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl, num_circuits) +twirled_circuits = [ + pt.twirl_CZ_gates(c, num_circuits=1)[0] for c in CNOT_twirled_circuits +] +print(CNOT_twirled_circuits) +print(twirled_circuits) +``` +As shown in the next section, this approach is exactly equivalent to the application of the standard Mitiq function {func}`.execute_with_pt()`. ## Executing the modified circuit From 3f596d2a79859a2f86a61c7ed70142f7de23bb6f Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 09:25:11 -0400 Subject: [PATCH 09/16] missed error in save --- docs/source/guide/pt-4-low-level.md | 1 - 1 file changed, 1 deletion(-) diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index 654cb92d3d..2a03f35001 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -50,7 +50,6 @@ In the next sections instead, we show how one can apply PT at a lower level, i.e First let's define our circuit: ```{code-cell} ipython3 from cirq import LineQubit, Circuit, CZ, CNOT -fr a, b, c, d = LineQubit.range(4) circuit = Circuit( From 2a79e4008b96b35fea35e3c7637fc89911f63171 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 14:20:45 -0400 Subject: [PATCH 10/16] clarified guide and made consistent --- docs/source/guide/pt-4-low-level.md | 51 ++++++++++++++++------------- docs/source/guide/pt-5-theory.md | 18 +++++----- docs/source/refs.bib | 9 +++++ 3 files changed, 47 insertions(+), 31 deletions(-) diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index 2a03f35001..7998603453 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -29,9 +29,9 @@ Workflow of the PT technique in Mitiq, detailed in the [What happens when I use - The error mitigated expectation value is returned to the user. With respect to the workflows of other error-mitigation techniques (e.g. [ZNE](zne-4-low-level.md) or [PEC](pec-4-low-level.md)), -PT involves the generation and the execution of a _single_ modified circuit. -For this reason, there is no need to combine the results of multiple circuits and the final inference step which is necessary for other -techniques is instead trivial for PT. +PT involves the generation of a _single_ randomly-modified circuit, and then averages over its executions. +For this reason, there is no need for a complex final inference step, which is necessary for other +techniques, and so this average is instead trivial for PT. ```{note} When setting the `num_trials` option to a value larger than one, multiple circuits are actually generated by Mitiq and @@ -45,8 +45,11 @@ In the next sections instead, we show how one can apply PT at a lower level, i.e - Twirling CZ and CNOT gates in the circuit; - Executing the modified circuit. +- Estimate expectation value by averaging over randomized twirling circuits ## Twirling CZ and CNOT gates in the circuit +To twirl about particular gates, we need the Pauli group for those gates. These groups are stored as lookup tables, in {attr}`mitiq.pt.pt.CNOT_twirling_gates` and {attr}`mitiq.pt.pt.CZ_twirling_gates`, so that we can randomly select a tuple from the group. Now we're ready to twirl our gates. + First let's define our circuit: ```{code-cell} ipython3 from cirq import LineQubit, Circuit, CZ, CNOT @@ -65,19 +68,21 @@ Now, we can see what happens when we apply the PT functions, through {func}`.twi from mitiq import pt circuit_to_twirl = circuit.copy() -CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl, num_circuits) +CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl) twirled_circuits = [ pt.twirl_CZ_gates(c, num_circuits=1)[0] for c in CNOT_twirled_circuits ] print(CNOT_twirled_circuits) print(twirled_circuits) ``` -As shown in the next section, this approach is exactly equivalent to the application of the standard Mitiq function {func}`.execute_with_pt()`. - -## Executing the modified circuit +We see that we return lists of the randomly twirled circuits, and so we must take a simple average over their expectation values. +## Executing the modified circuits +Now that we have our twirled circuits, let's simulate some noise and execute those circuits, using the {class}`mitiq.Executor` to collect the results. ```{code-cell} ipython3 from cirq import DensityMatrixSimulator, amplitude_damp +from mitiq import Executor + def execute(circuit, noise_level=0.003): """Returns Tr[ρ |00..⟩⟨00..|] where ρ is the state prepared by the circuit @@ -86,33 +91,35 @@ def execute(circuit, noise_level=0.003): noisy_circuit = circuit.with_noise(amplitude_damp(noise_level)) rho = DensityMatrixSimulator().simulate(noisy_circuit).final_density_matrix return rho[0, 0].real + +executor = Executor(execute) +expvals = executor.evaluate(twirled_circuits, None) ``` -If executed on a noiseless backend, `circuit_with_pt` and `circuit` are equivalent. -On a real backend, they have a different sensitivity to noise. The core idea of the PT technique is that, -`circuit_with_pt` is (hopefully) less sensitive to Markovian noise thanks to the randomized Pauli insertions. -```{code-cell} ipython3 -# Ideal result -execute(circuit, noise_level=0) -``` +## Estimate expectation value by averaging over randomized twirling circuits +Pauli Twirling doesn't require running the circuit at different noise levels or with different noise models. It applies a randomized sequence of Pauli operations within the same quantum circuit and averages the results to reduce the effect of the noise. ```{code-cell} ipython3 -# Unmitigated result -execute(circuit) -``` +import numpy as np -```{code-cell} ipython3 -# Error-mitigated result (with PT) -execute(circuit_with_pt) +average = cast(float, np.average(expvals)) +print(average) ``` +Keep in mind, ths code is for illustration and that the noise level, type of noise (here amplitude damping), and the observable need to be adapted to the specific experiment. + +If executed on a noiseless backend, a given `circuit_with_pt` and `circuit` are equivalent. +On a real backend, they have a different sensitivity to noise. The core idea of the PT technique is that, +`circuits_with_pt` (hopefully) tailors the noise into stochastic Pauli channels, such that a simple average over results +will return a mitigated result. + As a final remark, we stress that the low-level procedure that we have shown is exactly what {func}`.execute_with_pt()` does behind the scenes. Let's verify this fact: ```{code-cell} ipython3 np.isclose( - pt.execute_with_pt(circuit, execute), - execute(circuit_with_pt), + pt.execute_with_pt(circuit, executor), + average, ) ``` diff --git a/docs/source/guide/pt-5-theory.md b/docs/source/guide/pt-5-theory.md index dfc2599bcc..85efcce03e 100644 --- a/docs/source/guide/pt-5-theory.md +++ b/docs/source/guide/pt-5-theory.md @@ -13,7 +13,7 @@ kernelspec: # What is the theory behind Pauli Twirling? -Pauli Twirling (PT) {cite}`Wallman_2016_PRA, Hashim_2021_PRX, Urbanek_2021_PRL` +Pauli Twirling (PT) {cite}`Wallman_2016_PRA, Hashim_2021_PRX, Urbanek_2021_PRL, Saki_2023_arxiv` is a quantum error mitigation technique designed to tailor noise from a Markovian channel to a more manageable stochastic Pauli channel. This tailoring is achieved by randomly applying a series of Pauli operations to the quantum system, and in doing @@ -25,19 +25,19 @@ The application of DD sequences can have two effects depending on the correlatio 2. For non-Markovian noise, PT's performance might be degraded due to the inability to completely decouple the system from the environment in the presence of memory effects -Pauli Twirling (PT) can be a powerful tool for noise management in quantum systems. By twirling over the Pauli gates, PT transforms complex noise channels into simpler stochastic Pauli noise channels. Yet, as with all powerful tools, careful handling is required. +Pauli Twirling (PT) can be a powerful tool for noise management in quantum systems. By twirling over the Pauli gates, PT transforms complex noise channels into simpler stochastic Pauli noise channels. The success of PT is contingent on various factors, such as the nature of the noise and the specific characteristics of the quantum system. It's worth noting that, while PT generally simplifies the noise channel, there are circumstances where it could transform the noise negatively, for example into a completely depolarizing channel with a corresponding total loss of quantum information. For optimal results, Pauli Twirling should be implemented with an understanding of the underlying noise dynamics, and ideally, should be complemented with other error mitigation techniques to ensure robust quantum computation. -In the context of quantum error mitigation, PT and DDD stand apart in that they have very peculiar features with respect to other error mitigation techniques. PT's peculiarities include: +In the context of quantum error mitigation, PT is closer to [DDD](ddd-5-theory.md) in that they have very peculiar features with respect to other error mitigation techniques. PT's peculiarities include: -- It maps a noisy quantum computation to a _single_ error-mitigated computation (no need to take linear combinations -of noisy results as in [ZNE](zne-5-theory.md [PEC](pec-5-theory.md) and [CDR](cdr-5-theory.md)). +- It constructs a _single_ randomly-modified circuit, and subsequently averages over its executions. +With a single circuit, both the computational cost and complexity are reduced, making the final average of results a relatively straightforward task. That is, there is no need to take a linear combinations of noisy results as in [ZNE](zne-5-theory.md) [PEC](pec-5-theory.md) and [CDR](cdr-5-theory.md). -- As a consequence of the previous point, there is not a fundamental error mitigation overhead or -increase in statistical uncertainty in the final result. - -- If noise is time-correlated, PT should not be expected to yield positive (noise-reducing) results on its own +- As a consequence of the previous point, the fundamental error mitigation overhead is minimized, +such that there is no increase in statistical uncertainty in the final result, assuming optimal executions. +- If noise is time-correlated, PT should not be expected to yield positive (noise-reducing) results on its own, +though in this case composing with other techniques may yield the desired mitigation. diff --git a/docs/source/refs.bib b/docs/source/refs.bib index fbfc70cbc9..1be6b1e6a3 100644 --- a/docs/source/refs.bib +++ b/docs/source/refs.bib @@ -625,6 +625,15 @@ @article{Sagastizabal_2019_PRA url = {https://link.aps.org/doi/10.1103/PhysRevA.100.010302}, } +@misc{Saki_2023_arxiv, + title={Hypothesis Testing for Error Mitigation: How to Evaluate Error Mitigation}, + author={Abdullah Ash Saki and Amara Katabarwa and Salonik Resch and George Umbrarescu}, + year={2023}, + eprint={2301.02690}, + archivePrefix={arXiv}, + primaryClass={quant-ph} +} + @article{Schlosshauer_2019, doi = {10.1016/j.physrep.2019.10.001}, url = {https://doi.org/10.1016%2Fj.physrep.2019.10.001}, From 11db8062acfe9a88adf8a6047a0d2b599d28f5f5 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 14:53:58 -0400 Subject: [PATCH 11/16] missed save --- docs/source/guide/pt-4-low-level.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index 7998603453..216a2cabd8 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -68,7 +68,7 @@ Now, we can see what happens when we apply the PT functions, through {func}`.twi from mitiq import pt circuit_to_twirl = circuit.copy() -CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl) +CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl, num_circuits=10) twirled_circuits = [ pt.twirl_CZ_gates(c, num_circuits=1)[0] for c in CNOT_twirled_circuits ] From 4447d234074d0e9d55f03a40c9fbc8bc66be4250 Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 18:41:29 -0400 Subject: [PATCH 12/16] fix user guide --- docs/source/guide/pt-4-low-level.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index 216a2cabd8..8b2dc37bcf 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -72,8 +72,8 @@ CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl, num_circuits=10) twirled_circuits = [ pt.twirl_CZ_gates(c, num_circuits=1)[0] for c in CNOT_twirled_circuits ] -print(CNOT_twirled_circuits) -print(twirled_circuits) +print(CNOT_twirled_circuits[0:1]) +print(twirled_circuits[0:1]) ``` We see that we return lists of the randomly twirled circuits, and so we must take a simple average over their expectation values. @@ -102,6 +102,7 @@ Pauli Twirling doesn't require running the circuit at different noise levels or ```{code-cell} ipython3 import numpy as np +from typing import cast average = cast(float, np.average(expvals)) print(average) From 9017b024cd77d7bf3b55d3c4f9825de6729f8d1e Mon Sep 17 00:00:00 2001 From: Aaron Robertson Date: Tue, 30 May 2023 20:34:24 -0400 Subject: [PATCH 13/16] clean up workflow docs and rename image --- docs/source/guide/pt-4-low-level.md | 2 +- docs/source/img/pt_workflow.svg | 6 +++--- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index 8b2dc37bcf..eca3d1da61 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -18,7 +18,7 @@ The workflow of Pauli Twirling (PT) in Mitiq is represented in the figure below. ```{figure} ../img/pt_workflow.svg --- width: 700px -name: pt-workflow-overview +name: pt-workflow-overview-2 --- Workflow of the PT technique in Mitiq, detailed in the [What happens when I use PT?](pt-4-low-level.md) section. ``` diff --git a/docs/source/img/pt_workflow.svg b/docs/source/img/pt_workflow.svg index 7d1484685c..b045dc8acf 100644 --- a/docs/source/img/pt_workflow.svg +++ b/docs/source/img/pt_workflow.svg @@ -6,7 +6,7 @@ version="1.1" id="svg993" inkscape:version="1.2.2 (b0a8486, 2022-12-01)" - sodipodi:docname="ddd_workflow.svg" + sodipodi:docname="pt_workflow.svg" inkscape:export-filename="G:\My Drive\collaborations\zne_workflow2_steps.png" inkscape:export-xdpi="600" inkscape:export-ydpi="600" @@ -113,7 +113,7 @@ id="text1727" x="-40.748028" y="334.24435" - style="font-weight:400;font-stretch:semi-condensed;font-size:16.9333px;font-family:'Lucida Console', 'Lucida Console_MSFontService', sans-serif;stroke-width:0.264583">mitiq.ddd + style="font-weight:400;font-stretch:semi-condensed;font-size:16.9333px;font-family:'Lucida Console', 'Lucida Console_MSFontService', sans-serif;stroke-width:0.264583">mitiq.pt around noisy gates + id="tspan924">on CNOT/CZ Date: Fri, 2 Jun 2023 05:47:31 -0400 Subject: [PATCH 14/16] make clearer pt expected effect --- docs/source/guide/pt-1-intro.md | 15 ++++++++------- docs/source/guide/pt-2-use-case.md | 6 ++---- docs/source/guide/pt-4-low-level.md | 2 +- docs/source/guide/pt-5-theory.md | 25 +++++++++++-------------- docs/source/guide/pt.md | 7 +++---- 5 files changed, 25 insertions(+), 30 deletions(-) diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md index c1990b654c..c7d7917699 100644 --- a/docs/source/guide/pt-1-intro.md +++ b/docs/source/guide/pt-1-intro.md @@ -23,8 +23,9 @@ mitiq.SUPPORTED_PROGRAM_TYPES.keys() ## Problem setup -We first define the circuit of interest. In this example, the circuit has a -slack window with a length of 4 (in the sense that 4 single-qubit gates can fit in that window). +We first define the circuit of interest. In this example, the circuit has +two CNOT gates and a CZ gate. We can see that when we apply Pauli Twirling, +we will generate ```{code-cell} ipython3 from cirq import LineQubit, Circuit, CZ, CNOT @@ -90,11 +91,11 @@ Here we observe that the application of PT reduces the estimation error when com to the unmitigated result. ```{admonition} Note: -PT is designed to mitigate the simple Markovian noise simulated in this example, -but PT can still have a non-trivial effect on the final error, but it is not -always a positive effect. For example, one can check that by changing the -parameters of the input circuit, the error with PT is sometimes larger than -the unmitigated error. +PT is designed to transform the noise simulated in this example, +but it should nnot be expected to always be a positive effect. +In this sense, it is more of a noise tailoring technique, designed +to be composed with other techniques rather than an error mitigation +technique in and of itself. ``` +++ diff --git a/docs/source/guide/pt-2-use-case.md b/docs/source/guide/pt-2-use-case.md index 707ac07961..905dd3a2ae 100644 --- a/docs/source/guide/pt-2-use-case.md +++ b/docs/source/guide/pt-2-use-case.md @@ -15,7 +15,7 @@ kernelspec: ## Advantages -Pauli Twirling is a technique devised to minimize Markovian noise, by tailoring it towards Pauli channels. +Pauli Twirling is a technique devised to tailor noise towards Pauli channels. More details on the theory of Pauli Twirling are given in the section [What is the theory behind PT?](pt-5-theory.md). @@ -27,6 +27,4 @@ Pauli Twirling is agnostic to our knowledge on the type of noise, easy to implem Pauli Twirling is generally combined with a compilation pass to maintain circuit depth. Mitiq thus far does not provide this compilation, and so circuit depth is increased by the additional single-qubit gates. -Though the noise is tailored towards a more mitigable channel, it's possible for this channel to be entirely noisy (i.e. a completely depolarizing channel). - -Furthermore, though there are extensions to account for memory effects, the basic implementation in Mitiq should not be expected to mitigate non-Markovian noise (on its own). +Though the noise is tailored towards a more mitigable channel, it's possible for this channel to be entirely noisy (i.e. a completely depolarizing channel). In this way, it should not be expected to reduce noise on its own. diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index eca3d1da61..b5f068fb5c 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -29,7 +29,7 @@ Workflow of the PT technique in Mitiq, detailed in the [What happens when I use - The error mitigated expectation value is returned to the user. With respect to the workflows of other error-mitigation techniques (e.g. [ZNE](zne-4-low-level.md) or [PEC](pec-4-low-level.md)), -PT involves the generation of a _single_ randomly-modified circuit, and then averages over its executions. +PT involves the generation of a _single_ circuit with random modifications, and subsequently averages over many executions. For this reason, there is no need for a complex final inference step, which is necessary for other techniques, and so this average is instead trivial for PT. diff --git a/docs/source/guide/pt-5-theory.md b/docs/source/guide/pt-5-theory.md index 85efcce03e..b138590b9a 100644 --- a/docs/source/guide/pt-5-theory.md +++ b/docs/source/guide/pt-5-theory.md @@ -14,30 +14,27 @@ kernelspec: # What is the theory behind Pauli Twirling? Pauli Twirling (PT) {cite}`Wallman_2016_PRA, Hashim_2021_PRX, Urbanek_2021_PRL, Saki_2023_arxiv` -is a quantum error mitigation technique designed to tailor noise from a Markovian -channel to a more manageable stochastic Pauli channel. This tailoring is achieved -by randomly applying a series of Pauli operations to the quantum system, and in doing -so reduces the complexity of the noise channel. +is a quantum noise tailoring technique designed to transform the noise channel +towards a more manageable stochastic Pauli channel. This tailoring is achieved +by randomly applying a series of Pauli operations to the quantum system, then +averaging over the results, and in doing so can reduce the complexity of the errors. -The application of DD sequences can have two effects depending on the correlation time {cite}`Breuer_2007_Oxford` of the environment: +1. In general, PT is a noise agnostic tailoring technique, designed to be composed with more direct mitigation -1. For Markovian noise, PT can make the overall quantum channel more symmetric (analogous to dynamical decoupling {cite}`Viola_1998_PRA, Viola_1999_PRL, Zhang_2014_PRL`) - -2. For non-Markovian noise, PT's performance might be degraded due to the inability to completely decouple the system from the environment in the presence of memory effects +2. For Markovian noise, PT can make the overall quantum channel more symmetric (analogous to dynamical decoupling {cite}`Viola_1998_PRA, Viola_1999_PRL, Zhang_2014_PRL`) Pauli Twirling (PT) can be a powerful tool for noise management in quantum systems. By twirling over the Pauli gates, PT transforms complex noise channels into simpler stochastic Pauli noise channels. The success of PT is contingent on various factors, such as the nature of the noise and the specific characteristics of the quantum system. It's worth noting that, while PT generally simplifies the noise channel, there are circumstances where it could transform the noise negatively, for example into a completely depolarizing channel with a corresponding total loss of quantum information. -For optimal results, Pauli Twirling should be implemented with an understanding of the underlying noise dynamics, and ideally, should be complemented with other error mitigation techniques to ensure robust quantum computation. +For optimal results, Pauli Twirling should be implemented with an understanding of the underlying noise dynamics, and ideally, should be complemented with more direction error mitigation techniques to ensure robust quantum computation. + +In the context of quantum error mitigation, PT is closer to [DDD](ddd-5-theory.md), but stands apart as a noise tailoring technique. PT's peculiarities include: -In the context of quantum error mitigation, PT is closer to [DDD](ddd-5-theory.md) in that they have very peculiar features with respect to other error mitigation techniques. PT's peculiarities include: +- It is not expected to reduce noise on its own, but rather tailor the noise such that it can be properly mitigated by other techniques. -- It constructs a _single_ randomly-modified circuit, and subsequently averages over its executions. +- It constructs a _single_ circuit with random modifications, and subsequently averages over many executions. With a single circuit, both the computational cost and complexity are reduced, making the final average of results a relatively straightforward task. That is, there is no need to take a linear combinations of noisy results as in [ZNE](zne-5-theory.md) [PEC](pec-5-theory.md) and [CDR](cdr-5-theory.md). - As a consequence of the previous point, the fundamental error mitigation overhead is minimized, such that there is no increase in statistical uncertainty in the final result, assuming optimal executions. - -- If noise is time-correlated, PT should not be expected to yield positive (noise-reducing) results on its own, -though in this case composing with other techniques may yield the desired mitigation. diff --git a/docs/source/guide/pt.md b/docs/source/guide/pt.md index 20cf78858c..fe824a010d 100644 --- a/docs/source/guide/pt.md +++ b/docs/source/guide/pt.md @@ -1,11 +1,10 @@ # Pauli Twirling -Pauli Twirling (PT) is an error mitigation technique in which, +Pauli Twirling (PT) is an noise tailoring technique in which, in the Mitiq implementation, particularly noisy operations (e.g. CZ and CNOT) -are mitigated by independent, random, single-qubit gates inserted into +are transformed by independent, random, single-qubit gates inserted into the circuit such that the effective logical circuit remains unchanged -but the noise is tailored into stochastic Pauli errors, as opposed to -the Markovian processes that are otherwise dominant. +but the noise is tailored towards stochastic Pauli errors. For more discussion of the theory of PT, see the section [What is the theory behind PT?](pt-5-theory.md). From 3a8dd9ec933f13be4b207cfb568c11486c502e72 Mon Sep 17 00:00:00 2001 From: Purva Thakre Date: Fri, 29 Sep 2023 16:18:04 -0500 Subject: [PATCH 15/16] validate failure --- mitiq/pt/pt.py | 1 - mitiq/pt/tests/test_pt.py | 2 -- 2 files changed, 3 deletions(-) diff --git a/mitiq/pt/pt.py b/mitiq/pt/pt.py index 26c6d6b0de..c7c07f3902 100644 --- a/mitiq/pt/pt.py +++ b/mitiq/pt/pt.py @@ -50,7 +50,6 @@ ] - def pauli_twirl_circuit( circuit: QPROGRAM, num_circuits: int = 10, diff --git a/mitiq/pt/tests/test_pt.py b/mitiq/pt/tests/test_pt.py index 9fd6e118aa..e707d3b65b 100644 --- a/mitiq/pt/tests/test_pt.py +++ b/mitiq/pt/tests/test_pt.py @@ -118,7 +118,6 @@ def test_twirl_CNOT_increases_layer_count(): assert num_gates_after == num_gates_before - def test_pauli_twirl_circuit(): num_qubits = 3 num_layers = 20 @@ -145,4 +144,3 @@ def test_no_CNOT_CZ_circuit(twirl_func): for i in range(5): assert _equal(circuit, twirled_output[i]) - From ff51d9f3c0709e90b6f6b1e75a903c68d8114160 Mon Sep 17 00:00:00 2001 From: Purva Thakre Date: Fri, 29 Sep 2023 17:26:51 -0500 Subject: [PATCH 16/16] revert back + admonition --- docs/source/guide/pt-1-intro.md | 18 +- docs/source/guide/pt-2-use-case.md | 5 + docs/source/guide/pt-3-options.md | 5 + docs/source/guide/pt-4-low-level.md | 41 ++-- docs/source/guide/pt-5-theory.md | 5 + docs/source/guide/pt.md | 5 + docs/source/img/pt_workflow.png | Bin 0 -> 923815 bytes docs/source/img/pt_workflow.svg | 331 +++++++++++++++++++--------- docs/source/refs.bib | 2 +- mitiq/pt/pt.py | 2 +- 10 files changed, 286 insertions(+), 128 deletions(-) create mode 100644 docs/source/img/pt_workflow.png diff --git a/docs/source/guide/pt-1-intro.md b/docs/source/guide/pt-1-intro.md index c7d7917699..9a384a1d2e 100644 --- a/docs/source/guide/pt-1-intro.md +++ b/docs/source/guide/pt-1-intro.md @@ -13,6 +13,11 @@ kernelspec: # How do I use PT? +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. This users guide will change in the future +after some utility functions are introduced. +``` + As with all techniques, PT is compatible with any frontend supported by Mitiq: ```{code-cell} ipython3 @@ -73,26 +78,25 @@ print(f"Error without mitigation: {abs(ideal_value - noisy_value) :.3}") ## Apply PT Pauli Twirling can be easily implemented with the function -{func}`.execute_with_pt()`. +{func}`.pauli_twirl_circuit()`. ```{code-cell} ipython3 from mitiq import pt -mitigated_result = pt.execute_with_pt( +mitigated_result = pt.pauli_twirl_circuit( circuit=circuit, - executor=execute, ) ``` ```{code-cell} ipython3 -print(f"Error with mitigation (PT): {abs(ideal_value - mitigated_result) :.3}") +# print(f"Error with mitigation (PT): {abs(ideal_value - mitigated_result) :.3}") ``` -Here we observe that the application of PT reduces the estimation error when compared -to the unmitigated result. +Here we observe that the application of PT does not reduce the estimation error when compared +to the unmitigated result. The intended effect was to only tailor the noise. ```{admonition} Note: PT is designed to transform the noise simulated in this example, -but it should nnot be expected to always be a positive effect. +but it should not be expected to always be a positive effect. In this sense, it is more of a noise tailoring technique, designed to be composed with other techniques rather than an error mitigation technique in and of itself. diff --git a/docs/source/guide/pt-2-use-case.md b/docs/source/guide/pt-2-use-case.md index 905dd3a2ae..0c10da3ea4 100644 --- a/docs/source/guide/pt-2-use-case.md +++ b/docs/source/guide/pt-2-use-case.md @@ -13,6 +13,11 @@ kernelspec: # When should I use PT? +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. This users guide will change in the future +after some utility functions are introduced. +``` + ## Advantages Pauli Twirling is a technique devised to tailor noise towards Pauli channels. diff --git a/docs/source/guide/pt-3-options.md b/docs/source/guide/pt-3-options.md index e0b944dede..1d2091661f 100644 --- a/docs/source/guide/pt-3-options.md +++ b/docs/source/guide/pt-3-options.md @@ -13,4 +13,9 @@ kernelspec: # What additional options are available when using PT? +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. This users guide will change in the future +after some utility functions are introduced. +``` + Currently Pauli Twirling is designed to have relatively few options, in part to make it readily composable with every other Mitiq technique. In the future, we expect a possibility of supporting additional operations as targets (beyond CZ and CNOT gates), with more customization on picking those targets. Stay tuned! \ No newline at end of file diff --git a/docs/source/guide/pt-4-low-level.md b/docs/source/guide/pt-4-low-level.md index b5f068fb5c..14d6f2a650 100644 --- a/docs/source/guide/pt-4-low-level.md +++ b/docs/source/guide/pt-4-low-level.md @@ -13,6 +13,11 @@ kernelspec: # What happens when I use PT? +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. This users guide will change in the future +after some utility functions are introduced. +``` + The workflow of Pauli Twirling (PT) in Mitiq is represented in the figure below. ```{figure} ../img/pt_workflow.svg @@ -39,12 +44,12 @@ the associated results are averaged to obtain the final expectation value. This it can be considered as an average of independent single-circuit workflows. ``` -As shown in [How do I use PT?](pt-1-intro.md), the function {func}`.execute_with_pt()` applies PT behind the scenes -and directly returns the error-mitigated expectation value. +As shown in [How do I use PT?](pt-1-intro.md), the function {func}`.pauli_twirl_circuit()` applies PT behind the scenes +and returns the different versions of Pauli Twirled circuits. In the next sections instead, we show how one can apply PT at a lower level, i.e., by: - Twirling CZ and CNOT gates in the circuit; -- Executing the modified circuit. +- Executing the modified circuit (still under construction). - Estimate expectation value by averaging over randomized twirling circuits ## Twirling CZ and CNOT gates in the circuit @@ -72,12 +77,18 @@ CNOT_twirled_circuits = pt.twirl_CNOT_gates(circuit_to_twirl, num_circuits=10) twirled_circuits = [ pt.twirl_CZ_gates(c, num_circuits=1)[0] for c in CNOT_twirled_circuits ] -print(CNOT_twirled_circuits[0:1]) -print(twirled_circuits[0:1]) +print("Twirling just the CNOT gates: \n", CNOT_twirled_circuits[0], "\n") +print("Twirling both CNOT and CZ gates: \n" ,twirled_circuits[0]) ``` We see that we return lists of the randomly twirled circuits, and so we must take a simple average over their expectation values. ## Executing the modified circuits + +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. Some lines in the code blocks below are commented out as intended behavior +is currently a WIP. +``` + Now that we have our twirled circuits, let's simulate some noise and execute those circuits, using the {class}`mitiq.Executor` to collect the results. ```{code-cell} ipython3 from cirq import DensityMatrixSimulator, amplitude_damp @@ -93,7 +104,7 @@ def execute(circuit, noise_level=0.003): return rho[0, 0].real executor = Executor(execute) -expvals = executor.evaluate(twirled_circuits, None) +# expvals = executor.evaluate(twirled_circuits, None) ``` @@ -101,11 +112,11 @@ expvals = executor.evaluate(twirled_circuits, None) Pauli Twirling doesn't require running the circuit at different noise levels or with different noise models. It applies a randomized sequence of Pauli operations within the same quantum circuit and averages the results to reduce the effect of the noise. ```{code-cell} ipython3 -import numpy as np -from typing import cast +# import numpy as np +# from typing import cast -average = cast(float, np.average(expvals)) -print(average) +# average = cast(float, np.average(expvals)) +# print(average) ``` Keep in mind, ths code is for illustration and that the noise level, type of noise (here amplitude damping), and the observable need to be adapted to the specific experiment. @@ -115,12 +126,12 @@ On a real backend, they have a different sensitivity to noise. The core idea of `circuits_with_pt` (hopefully) tailors the noise into stochastic Pauli channels, such that a simple average over results will return a mitigated result. -As a final remark, we stress that the low-level procedure that we have shown is exactly what {func}`.execute_with_pt()` does behind the scenes. +As a final remark, we stress that the low-level procedure that we have shown is exactly what {func}`.pauli_twirl_circuit()` does behind the scenes. Let's verify this fact: ```{code-cell} ipython3 -np.isclose( - pt.execute_with_pt(circuit, executor), - average, -) +# np.isclose( +# pt.pauli_twirl_circuit(circuit, executor), +# average, +# ) ``` diff --git a/docs/source/guide/pt-5-theory.md b/docs/source/guide/pt-5-theory.md index b138590b9a..653630be37 100644 --- a/docs/source/guide/pt-5-theory.md +++ b/docs/source/guide/pt-5-theory.md @@ -13,6 +13,11 @@ kernelspec: # What is the theory behind Pauli Twirling? +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. This users guide will change in the future +after some utility functions are introduced. +``` + Pauli Twirling (PT) {cite}`Wallman_2016_PRA, Hashim_2021_PRX, Urbanek_2021_PRL, Saki_2023_arxiv` is a quantum noise tailoring technique designed to transform the noise channel towards a more manageable stochastic Pauli channel. This tailoring is achieved diff --git a/docs/source/guide/pt.md b/docs/source/guide/pt.md index fe824a010d..82f3965efb 100644 --- a/docs/source/guide/pt.md +++ b/docs/source/guide/pt.md @@ -1,5 +1,10 @@ # Pauli Twirling +```{admonition} Warning: +Pauli Twirling in Mitiq is still under construction. 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z-}{TXHyaCH@m)7-zH9ul#J5(U3#n~B-JhzR(RfIA=S&b>GI8~JYY8V(Jy!gKQn$P- zmi(W3cKbT2^~(_EvIoz$T{lub&p6_se3|;*U4OCeD|Lp~eaP~w=-*nxCM&nZmgNyn d3OgBys!f?S`wVClYy4Mtz1M%}S?Eqr{}+#3@&*6^ literal 0 HcmV?d00001 diff --git a/docs/source/img/pt_workflow.svg b/docs/source/img/pt_workflow.svg index b045dc8acf..7b8dfb1d81 100644 --- a/docs/source/img/pt_workflow.svg +++ b/docs/source/img/pt_workflow.svg @@ -5,7 +5,7 @@ viewBox="0 0 438.94376 222.25" version="1.1" id="svg993" - inkscape:version="1.2.2 (b0a8486, 2022-12-01)" + inkscape:version="1.3 (1:1.3+202307231459+0e150ed6c4)" sodipodi:docname="pt_workflow.svg" inkscape:export-filename="G:\My Drive\collaborations\zne_workflow2_steps.png" inkscape:export-xdpi="600" @@ -19,6 +19,18 @@ xmlns:dc="http://purl.org/dc/elements/1.1/"> + + User + style="fill:#a2d0aa;fill-opacity:1;stroke-width:0.251429" /> + + + + 2. Execute each twirled circuit on backend + mitiq.pt + - + style="fill:#ffffe1;stroke:#000000;stroke-width:1.19062;stroke-miterlimit:8" /> Hardware + x="105.44978" + y="337.41998" + style="stroke-width:0.264583">Hardware + + + - + transform="matrix(0.82993196,0,0,0.82993196,-12.877404,241.97582)"> + style="fill:#f2f2f2;stroke-width:0.402073" /> 2. Inference (trivial for PT) + style="font-weight:400;font-stretch:semi-condensed;font-size:10.5833px;font-family:'Lucida Console', 'Lucida Console_MSFontService', sans-serif;white-space:pre;inline-size:158.949;stroke-width:0.264583" + transform="matrix(1.0939119,0,0,1,24.104172,-2.1153661)" + xml:space="preserve">3. Inference : Average over executed results + transform="matrix(1.0640995,0,0,1,2.000506,269.7649)"> + style="stroke-width:0.281367" /> 1. Generate circuit(s) + style="font-weight:400;font-stretch:semi-condensed;font-size:9.525px;font-family:'Lucida Console', 'Lucida Console_MSFontService', sans-serif;white-space:pre;inline-size:153.678;stroke-width:0.264583" + xml:space="preserve">1. Generate twirled circuit(s) - Circuit withtwirling gateson CNOT/CZ Error-mitigatedexpectationvalue Execute onbackend + x="186.78662" + y="358.97006" + style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9.87778px;line-height:1.25;font-family:'Lucida Console', 'Lucida Console_MSFontService', sans-serif;-inkscape-font-specification:'Lucida Console, Lucida Console_MSFontService, sans-serif';text-align:center;text-anchor:middle;stroke-width:0.358811" + id="tspan1013">Execute program + + + + Circuit withtwirling gateson CNOT/CZ + + diff --git a/docs/source/refs.bib b/docs/source/refs.bib index 11044ec0c2..1e4facb3f1 100644 --- a/docs/source/refs.bib +++ b/docs/source/refs.bib @@ -725,7 +725,7 @@ @article{Sagastizabal_2019_PRA } -@misc{saki2023hypothesis, +@misc{Saki_2023_arxiv, title={Hypothesis Testing for Error Mitigation: How to Evaluate Error Mitigation}, author={Abdullah Ash Saki and Amara Katabarwa and Salonik Resch and George Umbrarescu}, year={2023}, diff --git a/mitiq/pt/pt.py b/mitiq/pt/pt.py index c7c07f3902..a76598b559 100644 --- a/mitiq/pt/pt.py +++ b/mitiq/pt/pt.py @@ -58,7 +58,7 @@ def pauli_twirl_circuit( Only the $\mathrm{CZ}$ and $\mathrm{CNOT}$ gates in an input circuit are Pauli twirled as specified in - :cite:`saki2023hypothesis`. + :cite:`Saki_2023_arxiv`. Args: circuit: The input circuit to execute with twirling.