Variational Approach to Quantum State Discrimination

Abstract

Quantum state discrimination is an important task for quantum information processing. It is known that in quantum mechanics, one could only perfectly discriminate orthogonal states. When the states are non-orthogonal, several strategies can be adopted to discriminate them. One such strategy is the minimum error strategy where we minimize the probability of error resulting from imperfect discriminations. However, exact solution is only known for two states discrimination case. In this project, we are expecting a search based algorithm to help find the optimal quantum circuit for state discrimination.

Optimization Algorithm

• Local search based: simple hill climbing We use step by step approach of approximation to reach the top of the hill in success rate.

Result

Discrimination results

• The quantum states in forms of (a,b) in (a * |0> + b * |1>) qp_1: (1, 0)
  qp_2: (0.7071067811865476, 0.7071067811865476)
  qp_3: (0, 1)
  qp_4: (0.7071067811865476, 0.7071067811865476j)
  qp_5: (0.5, 0.8660254037844386)
  qp_6: (0.8660254037844386, 0.5)
  qp_7: (0.5773502691896257, 0.816496580927726)
  qp_8: (0.816496580927726, 0.5773502691896257)

• Discrimination success rates
  indexing: (1)q-(2) qp_(3) qp_(4) --> (1): number of qubits, (2): experiment index, (3),(4): state index

  2q-1 qp_1 qp_2 (Theoretical maximum is about 85%)
  average is: 0.8289742857142854
  stddev is: 0.01667364264891369

  2q-2 qp_1 qp_3
  average is: 0.9775971428571428
  stddev is: 0.018997885274365647

  2q-3 qp_7 qp_8
  average is: 0.6590907142857144
  stddev is: 0.008105584211404091

  2q-4 qp_5 qp_6
  average is: 0.7422142857142857
  stddev is: 0.009714390755734603

  3q-1 qp_1 qp_2 qp_3 (Theoretical maximum is about 67%)
  average is: 0.6167666666666666
  stddev is: 0.03715883325521523

  3q-2 qp_1 qp_2 qp_4
  average is: 0.5366933333333334
  stddev is: 0.0192063461734223

  3q-3 qp_1 qp_5 qp_6
  average is: 0.5426266666666667
  stddev is: 0.046376988546763864