[Paper Review] Equivalence determination of unitary operations

Atsushi Shimbo, Akihito Soeda+1 more|arXiv (Cornell University)|Mar 30, 2018

Quantum Computing Algorithms and Architecture2 references3 citations

TL;DR

This paper studies equivalence determination of unitary operations, where a target black-box unitary must be identified among two candidate unitaries using finite quantum samples. By formulating the task as a semidefinite program (SDP), the authors show that general adaptive protocols outperform parallelized schemes, though for specific sample counts, parallel schemes achieve optimal performance. Notably, finite quantum samples can enable equivalence determination as accurately as with full classical descriptions, despite the impossibility of obtaining exact classical descriptions from finite samples.

ABSTRACT

We study equivalence determination of unitary operations, a task analogous to quantum state discrimination. The candidate states are replaced by unitary operations given as a quantum sample, i.e., a black-box device implementing a candidate unitary operation, and the discrimination target becomes another black-box. The task is an instance of higher-order quantum computation with the black-boxes as input. The optimal error probability is calculated by semidefinite programs. Arbitrary quantum operations applied between the black-boxes in a general protocol provide advantages over protocols restricted to parallelized use of the black-boxes. We provide a numerical proof of such an advantage. In contrast, a parallelized scheme is analytically shown to exhibit the optimal performance of general schemes for a particular number of quantum samples of the candidates. We find examples of finite-sample equivalence determination that achieve the same performance as when a classical description of the candidates are provided, although an exact classical description cannot be obtained from finite quantum samples.

Motivation & Objective

• To investigate the task of determining whether a target black-box unitary matches one of two candidate unitaries provided as quantum samples.
• To analyze the performance advantage of general adaptive protocols over restricted parallelized protocols in unitary equivalence determination.
• To determine under what conditions finite quantum samples can achieve the same performance as full classical descriptions of the candidates.
• To characterize the optimal success probability using semidefinite programming under various concurrency patterns of quantum operation access.

Proposed method

• Formulates the equivalence determination task as a minimum-error discrimination problem using semidefinite programming (SDP) to compute optimal success probabilities.
• Models the protocol as a general quantum circuit with arbitrary quantum operations between black-box uses, allowing adaptive strategies.
• Uses representation theory of SU(2) to decompose the Hilbert space into multiplicity subspaces, enabling block-diagonalization of the SDP constraints.
• Derives SDP formulations for four distinct concurrency patterns, each corresponding to different access orders of the candidate and target black-boxes.
• Imposes symmetry constraints using irreducible representations to reduce the dimensionality of the optimization problem.
• Numerically evaluates the SDP solutions to demonstrate performance advantages of general protocols over parallel schemes.

Experimental results

Research questions

• RQ1 Can general adaptive protocols outperform parallelized protocols in unitary equivalence determination, and if so, by how much?
• RQ2 Under what conditions does a parallelized protocol achieve the same performance as a general adaptive protocol?
• RQ3 Can finite quantum samples of unitary operations enable equivalence determination with the same success probability as if a full classical description were available?
• RQ4 What is the optimal success probability for equivalence determination when only a finite number of samples of the candidate unitaries are available?
• RQ5 How does the structure of the Hilbert space decomposition via SU(2) representations affect the solution of the equivalence determination task?

Key findings

• General adaptive protocols provide a performance advantage over parallelized schemes in equivalence determination of unitary operations, as confirmed by numerical SDP solutions.
• For specific sample counts—particularly (1,1) and (2,1) configurations—parallelized schemes achieve the same optimal success probability as general protocols.
• Finite quantum samples can enable equivalence determination with the same performance as if a full classical description of the unitaries were known, despite the impossibility of reconstructing the classical description from finite samples.
• The optimal success probability for two-qubit unitary equivalence determination is analytically derived for certain configurations, showing agreement between parallel and general schemes.
• The SDP formulation reveals that the optimal strategy depends critically on the concurrency pattern, with distinct constraints arising from different access orders of the black-boxes.
• The use of SU(2) representation theory allows efficient block-diagonalization of the SDP, enabling numerical and analytical analysis of the equivalence task.