Efficient verification of quantum gates with local operations

Huangjun Zhu; Haoyu Zhang

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Efficient verification of quantum gates with local operations

Huangjun Zhu, Haoyu Zhang

arXiv:1910.14032v2Oct 30, 2019quant-ph

TL;DR

Scalable quantum-device verification is difficult because process tomography incurs exponential overhead and randomized benchmarking has restrictive applicability assumptions. The paper proposes local-operation verification for individual gates and gate sets, including circuits, and shows efficient verification across many important unitary families, often with system-size-independent sample complexity and little overhead from local Pauli operations.

Problem
Quantum-gate and circuit verification becomes resource-intensive with system size, while randomized benchmarking applies only under restrictive assumptions about the gate set.
Source: §I · p. 1
Method
The paper develops a general framework using local state preparation and measurements to verify individual unitary gates and gate sets, including circuits.
Source: §I · p. 1
Results
The framework verifies broad unitary families with sample complexity at most linear in the number of parties and often independent of system size.
Source: §I · p. 1, §VII · p. 7
Takeaways & Limitations
Efficient unitary verification remains possible using only local operations, including Pauli operations in many practical cases and scenarios where randomized benchmarking does not apply.
Source: §I · p. 1, §VII · p. 7

Abstract

from arXiv · show

Efficient verification of the functioning of quantum devices is a key to the development of quantum technologies, but is a daunting task as the system size increases. Here we propose a simple and general framework for verifying unitary transformations that can be applied to both individual quantum gates and gate sets, including quantum circuits. This framework enables efficient verification of many important unitary transformations, including but not limited to all bipartite unitaries, Clifford unitaries, generalized controlled-$Z$ gates, generalized controlled-NOT gates, the controlled-SWAP gate, and permutation transformations. For all these unitaries, the sample complexity increases at most linearly with the system size and is often independent of the system size. Moreover, little overhead is incurred even if one can only prepare Pauli eigenstates and perform local measurements. Our approach is applicable in many scenarios in which randomized benchmarking (RB) does not apply and is thus instrumental to quantum computation and many other applications in quantum information processing.

I. INTRODUCTION

Quantum-device verification becomes difficult at scale: process tomography has exponential resource overhead, while randomized benchmarking is efficient but depends on restrictive gate-set assumptions. The paper introduces a local-operation framework that verifies individual gates and gate sets, including circuits, across many unitary families.

Process tomography has resource overhead that increases exponentially with system size, making scalable gate and circuit characterization a bottleneck.Source: §I · p. 1
Randomized benchmarking is more efficient and robust to SPAM errors but generally assumes group structure and unitary 2-design properties.Source: §I · p. 1
The proposed framework uses local state preparation and measurements to verify individual gates and gate sets, including quantum circuits, even where RB does not apply.Source: §I · p. 1
It covers important families including bipartite, Clifford, generalized controlled-Z, generalized CNOT, controlled-SWAP, and permutation unitaries.Source: §I · p. 1

II. CHANNEL-STATE DUALITY

The paper represents quantum channels through Choi states, providing a state-based description of process behavior. This duality supports fidelity definitions and connects channel verification to state verification.

Average gate fidelity and entanglement fidelity compare a channel with a target unitary through channel action and Choi-state fidelity, respectively.Source: §II · pp. 1–2
Quantum channels on B(H) correspond one-to-one with Choi states on H ⊗ H [13].Source: §II · p. 1
The Choi state of a channel has a completely mixed second reduced state, and conversely this condition identifies a quantum channel.Source: §II · pp. 1–2

A. Basic framework

The basic framework randomly prepares test states, applies accessible two-outcome tests to the outputs, and aggregates them into verification operators. Spectral gaps quantify how well the resulting strategy distinguishes the target from imperfect processes.

The verifier samples pure input states and tests whether the channel output matches the corresponding target state, with the target guaranteed to pass.Source: A. Basic framework · p. 2
Weighted output tests define verification operators whose passing probabilities are averaged over the test-state ensemble.Source: A. Basic framework · p. 2
The process verification operator can be expressed in equivalent variants, with one form more natural and another more convenient for technical analysis.Source: A. Basic framework · p. 2
The measurement spectral gap is the minimum output-test gap, where each gap is one minus the second-largest eigenvalue of its verification operator.Source: A. Basic framework · p. 2

B. Number of tests required

The framework bounds false acceptance probabilities using entanglement or average gate infidelity and reduces their computation to semidefinite programs. It also establishes concavity and convexity properties that support test-complexity analysis.

The required number of tests is defined by raising the relevant passing-probability bound to the Nth power until it falls below δ, with the bound saturated when run infidelities are equal.Source: B. Number of tests required · pp. 2–3
The verification objective is to keep false acceptance below significance level δ whenever average gate or entanglement fidelity is below 1 − ǫ.Source: B. Number of tests required · pp. 2–3
The worst-case entanglement-fidelity passing probability can be computed by semidefinite programming under positivity, channel normalization, and fidelity constraints.Source: B. Number of tests required · p. 3
The acceptance functions are concave and nonincreasing in infidelity but convex in the verification operator.Source: B. Number of tests required · p. 3

C. Optimal verification strategies

The optimal verification strategy uses unitary symmetry and ideal output verification, with Haar-random test states or practical 2-design substitutes achieving the best supported efficiency.

Averaging over unitarily equivalent verification strategies cannot decrease efficiency, enabling symmetry-based optimization of the verification operator.Source: C. Optimal verification strategies · p. 3
Efficiency is maximized when each output verification operator equals the target output state and test states follow the Haar measure.Replacing Ω_j with U(ρ_j) cannot decrease efficiency under the stated assumption.Source: C. Optimal verification strategies · p. 3
The required test counts satisfy NE(ε,δ,Θ) = NA(dε/(d+1),δ,Θ), while pA(Θ,ε) ≥ 1−ε gives a universal lower bound on tests.Source: C. Optimal verification strategies · p. 3
Haar-random test states can be replaced by pseudo-random states from a 2-design while retaining the optimal verification relation.The 2-design option is easier to realize than Haar-random sampling.Source: C. Optimal verification strategies · p. 3
For any process verification operator, the stated lower bounds are saturated when test states form a 2-design and each output verification operator is the corresponding state projector.Source: C. Optimal verification strategies · p. 3

IV. BALANCED STRATEGIES

Balanced strategies normalize the test-state ensemble and reduce verification performance to spectral gaps associated with preparation and measurement. Their strongest preparation-gap bounds are attained by 2-designs, with mutually unbiased bases providing explicit constructions.

IV. BALANCED STRATEGIES

Balanced strategies use ensembles satisfying dΣ_j p_jρ_j = I, and the corresponding verification operators have the maximally entangled state as their largest-eigenvalue eigenstate.Uniform measurements over an orthonormal basis provide a simple balanced construction.Source: §IV · p. 4
The efficiency bounds for balanced strategies follow from the preparation and measurement gaps, including the relation ν ≥ νPνM used in the second upper bound.Source: §IV · p. 4
The strategy spectral gap is governed by preparation and measurement gaps, which characterize the performance of state preparation and measurements, respectively.For balanced strategies, ΘP ≤ Θ implies βP ≤ β and ν ≤ νP.Source: §IV · p. 4
For balanced strategies, ν ≤ νP ≤ d/(d+1), with the preparation bound saturated exactly by 2-design test-state ensembles; νM ≤ 1 is saturated exactly by state-projector verification operators.Source: §IV · p. 4
Using r mutually unbiased bases with uniform weights yields preparation gap νP = (r−1)/r.Source: §IV · p. 4
For prime dimension d, the eigenbases of Z, X, XZ, XZ^2, ..., XZ^(d−1) form a complete set of mutually unbiased bases.For prime-power dimensions, stabilizer states also form a 2-design, and complete sets of d+1 mutually unbiased bases provide another construction.Source: §IV · p. 4

V. LOCAL STATE PREPARATION AND MEASUREMENTS

The section characterizes how local state preparation and measurements constrain unitary verification, showing that product-state preparation can remain highly efficient while measurement limitations depend on the target transformation.

Optimal product strategies

Balanced product-state ensembles maximize preparation performance when distributed according to the normalized local Haar measure, and product 2-designs achieve the same bound.Source: §V · p. 5

Local state preparation

Product-state preparation has spectral gap at most dmin/(dmin + 1), with the bound attained by product 2-designs.Here dmin is the minimum local dimension.Source: §V · p. 5
Pauli-eigenstate constructions provide at least three mutually unbiased product bases, yielding preparation spectral gap at least 2/3.When local dimensions are prime powers, dmin mutually unbiased product bases attain dmin/(dmin + 1).Source: §V · p. 5
The restriction to local state preparation has little influence on verification efficiency.Source: §V · p. 5

Local measurements

Local-measurement limitations depend on both the unitary being verified and the chosen test states, although many gates and circuits remain efficiently verifiable.Source: §V · p. 5

VI. APPLICATIONS

The framework yields local-operation verification protocols for bipartite and multipartite gates, including Clifford, controlled, CSWAP, and permutation transformations.

Overview

The applications demonstrate efficient balanced verification protocols for important quantum gates and circuits using local operations.Source: §VI · p. 5

Bipartite unitaries

All bipartite unitaries can be verified efficiently with local preparation and measurements, using pure-state protocols with spectral gap at least 2/3.A two-test one-way-communication strategy reaches measurement spectral gap 1/2 with comparable test complexity.Source: §VI · p. 5

Clifford unitaries

Clifford gates and circuits require only local Pauli eigenstate preparation and Pauli measurements, with preparation spectral gap 2/3 for qubits.For nonprime local dimensions, the protocol remains efficient, and its test count is quadratically smaller than direct fidelity estimation.Source: §VI · pp. 5–6

Controlled gates

Generalized controlled-Z gates use two product bases and Pauli verification of hypergraph-state outputs, extending to controlled-X gates and IQP circuits.Source: §VI · p. 6

CSWAP gate

The CSWAP gate achieves measurement spectral gap 2/3 using three mutually unbiased product bases and Pauli-based verification of product or GHZ outputs.Source: §VI · p. 6

Permutation transformations

Permutation transformations attain process spectral gap d1/(d1 + 1) with product 2-designs, while two mutually unbiased bases require at most twice as many tests.Pauli eigenstates provide practical mutually unbiased product bases, including d1 + 1 bases when d1 is prime.Source: §VI · pp. 6–7

VII. SUMMARY

The framework verifies broad classes of unitary transformations with local operations, including gates and circuits, while retaining scalable efficiency.

Summary

The framework efficiently verifies bipartite, Clifford, controlled-Z, controlled-NOT, CSWAP, and permutation transformations using local state preparation and measurements.It applies to both individual gates and gate sets, including quantum circuits.Source: §VII · p. 7