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Noise Resilience of Variational Quantum Compiling

Kunal Sharma, Sumeet Khatri, M. Cerezo, Patrick J. Coles

arXiv:1908.04416v2quant-ph

TL;DR

NISQ quantum compiling must optimize short-depth gate sequences despite noise in quantum cost evaluations. This paper introduces Optimal Parameter Resilience and proves it for several VQC noise models, complementing simulations that recover optimal parameters across multiple compilation tasks. The results suggest VQC can remain useful for circuit-depth compression on noisy devices, while broader noise resilience remains an open direction.

  • Problem

    Quantum compiling seeks short low-level circuits, but NISQ cost evaluations are affected by decoherence, gate noise, and measurement noise.

  • Method

    The paper analyzes Optimal Parameter Resilience in full-unitary and fixed-input-state VQC through rigorous noise-model theorems and numerical simulations.

  • Results

    VQC optimal parameters remain resilient under measurement noise and several incoherent noise models, with IBM-simulator experiments covering quantum Fourier transform, Toffoli, and W-state preparation.

  • Takeaways & Limitations

    VQC’s noise resilience supports its use for circuit-depth compression on noisy intermediate-scale quantum devices.

Abstract

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Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary $U$ is compiled into a short-depth gate sequence $V$. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence $V$ (i.e., the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM's noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.

I. INTRODUCTION

Variational quantum compiling uses classical optimization to train short-depth circuits, and this work shows that optimal parameters can remain unchanged under several incoherent noise sources. The paper supports this claim analytically for VQC and illustrates broader potential relevance to VHQCAs.

  • NISQ devices suffer from decoherence, gate noise, and measurement noise, motivating error-mitigation strategies.
  • VHQCAs evaluate parameter-dependent cost functions quantumly while classical optimization updates the gate-sequence parameters.
  • Optimal Parameter Resilience means that global-optimum variational parameters remain unaffected by specified incoherent noise, even when the cost value may change.
  • VQC trains a short-depth sequence V(α) to approximate a target unitary U, supporting circuit-depth reduction for quantum algorithms.
  • Rigorous theorems establish resilience to measurement noise, incoherent gate noise, decoherence, Pauli channels, and non-unital Pauli channels during cost evaluation.
  • Numerical experiments on IBM’s noisy simulator observed resilience while compiling the quantum Fourier transform, Toffoli gate, and W-state preparation.

A. Full unitary matrix compiling

The paper develops full-unitary and fixed-input-state compiling formulations using global and local overlap-based cost functions. These formulations provide different trade-offs in generality, trainability, and the degeneracy of optimal solutions.

  • Full Unitary Matrix Compiling: The Hilbert-Schmidt cost is CHST = 1 − |Tr(V†U)|2/d2, with the HST estimating the corresponding overlap using a maximally entangled state.
  • Full Unitary Matrix Compiling: CHST is faithful: zero cost occurs exactly when V equals U up to a global phase.
  • Full Unitary Matrix Compiling: The local Hilbert-Schmidt cost is also faithful, while combining global and local costs balances operational meaning against trainability as system size grows.
  • Compiling with a fixed input state: Fixed Input State Compiling trains V to reproduce U’s action on a specified input, here the all-zero state, rather than compiling the entire unitary.
  • Compiling with a fixed input state: The Loschmidt Echo Test measures recovery of the input after U followed by V†, while the local variant measures only one qubit.
  • Compiling with a fixed input state: Fixed-input compiling admits more solutions: any optimal W can retain an arbitrary (n−1) × (n−1) principal submatrix, creating degenerate optima.

IV. NOISE PROCESSES

The paper models decoherence, incoherent gate noise, and measurement noise, then proves that full-unitary compiling retains the same optimal parameters under broad combinations of these processes.

  • Noise models: The noise framework covers decoherence, gate noise, and measurement noise, including T1, T2, cross-talk, and asymmetric readout-error cases.Pauli noise includes dephasing, non-unital Pauli noise includes amplitude damping, and Pauli gate noise allows global channels before and after gates.
  • Noise models: Measurement Noise replaces standard-basis POVM elements with noisy elements whose probabilities describe reported outcomes conditioned on input states.The model permits asymmetric readout errors while requiring correct outcomes to be more probable than incorrect ones.
  • Full-unitary compiling: Theorem 1 states that the HST and LHST cost functions exhibit strong-OPR under Noise Model 1.Noise Model 1 combines continuous depolarizing noise, timed Pauli noise, subsystem-specific depolarizing and non-unital Pauli noise, Pauli gate noise, and measurement noise.
  • Full-unitary compiling: Full-unitary compiling is resilient when global depolarizing, timed Pauli, measurement, Pauli gate, and subsystem-specific decoherence processes occur together.The allowed subsystem channels include dephasing through Pauli noise and amplitude damping through non-unital Pauli noise.
  • Full-unitary compiling: Theorem 2 and its corollaries extend strong-OPR to Noise Model 2 and specialized Pauli-channel processes during V†U.The extensions include non-unital Pauli noise at selected locations and conditions involving Clifford or tensor-product structure.

B. Noise Resilience of Fixed Input State Compiling

For fixed-input-state compiling, the LET and LLET costs retain weak optimal-parameter resilience under Noise Model 3 and several structured extensions. Noise can reduce degeneracy among optima, but remaining noisy optima are noiseless optima.

  • Noise Model 3: Noise Model 3 combines continuous global depolarizing noise, global Pauli noise at τ1, and measurement noise during LET or LLET evaluation.Because measurement follows V†U immediately, no post-V†U noisy channel is included.
  • Main result: Theorem 3 states that CLET and CLLET exhibit weak-OPR under Noise Model 3.The result applies to fixed-input-state compiling using the Loschmidt Echo Test and Local Loschmidt Echo Test.
  • Main result: Fixed-input-state compiling is resilient to measurement noise, Pauli noise at τ1, and continuous global depolarizing noise.These are the three processes explicitly included in Noise Model 3.
  • Interpretation: Noise may reduce the number of global optima in fixed-input-state compiling, while every remaining noisy optimum corresponds to a noiseless global optimum.Thus, weak-OPR reflects possible degeneracy breaking rather than weak resilience.
  • Extensions: Weak-OPR extensions permit structured Pauli channels during V†U when their overall action can be represented as the ideal gate sequence followed by a Pauli channel.Additional corollaries cover Clifford prefixes and tensor-product structure with local depolarizing channels.

VI. IMPLEMENTATIONS

The implementations test variational compiling on three-qubit Toffoli, QFT, and W-state circuits using IBM’s noisy simulator and hardware-informed noise models.

  • Benchmarks: The experiments compile three-qubit Toffoli, Quantum Fourier Transform, and W-state preparation unitaries.The selected circuits are associated with universal gate sets, Shor’s algorithm, and QAOA applications, respectively.
  • Benchmarks: The Toffoli circuit contains nine one-qubit gates and six CNOTs, while the QFT uses its textbook circuit and the W-state circuit comes from prior constructions.These gate sequences provide the target circuits for the implementation study.
  • Simulator and noise: The experiments use IBM’s noisy simulator with a noise model based on reported parameters and connectivity of the 14-qubit Melbourne computer.The simulator models single- and two-qubit gate errors with depolarizing and thermal-relaxation channels and includes readout errors.
  • Evaluation: The study compares noisy cost-function trajectories with noiseless costs evaluated at parameters learned during noisy optimization.This comparison tests whether noisy training recovers parameters that perform well under noiseless evaluation.

A. Ansatzes and optimization methods

The paper uses dressed-CNOT building blocks in alternating-pair and target-inspired ansatzes, optimized with gradient-based methods. Numerical results show noisy training can recover parameters minimizing noiseless costs, although ansatz completeness and depth affect compilation quality and trainability.

  • Ansatz construction: Dressed CNOTs combine a CNOT with parameterized single-qubit gates before and after it, providing the ansatz building block.Each single-qubit gate is parameterized by three rotation angles.
  • Ansatz construction: The alternating-pair ansatz uses layered dressed CNOTs acting on alternating pairs of neighboring qubits.Its structure is illustrated for four qubits and can become incomplete with few layers.
  • Ansatz trade-offs: Increasing the number of layers can improve compilation completeness but makes training harder and can produce longer-depth circuits.The alternating-pair ansatz may also introduce unnecessary depth compared with a target-dependent structure.
  • Ansatz construction: The target-inspired ansatz derives dressed-CNOT placements from the target unitary’s gate sequence and is always complete.It is useful for demonstrating optimal-parameter resilience, but does not compress the number of CNOTs.
  • Optimization: Gradient descent trains the gate parameters using cost-function gradients, with fixed-shot optimization for shallow ansatzes and iCANS for deeper ones.The shallow implementations use N = 50000 shots per iteration, while deeper cases use an adaptive shot strategy.
  • Numerical results: For Toffoli compilation, noisy training converges to the noise-free minima for one- and two-layer ansatzes, while the target-inspired case reaches costs of order 10^-4.The one-layer ansatz remains incomplete, and the reported minima may be local rather than global.
  • Numerical results: For the three-qubit QFT, noisy training reaches the noise-free reference in the one-layer case, while two-layer optimization terminates before one cost converges.The one-layer ansatz is incomplete, whereas the two-layer case has a CLHST dashed-line value of order 10^-4, implying completeness.

D. W-state preparation

The paper evaluates VQC under IBM’s noisy simulator and reports resilience across compiling tasks, while identifying boundaries and open questions for broader noise resilience.

  • W-state preparation: All four cost functions CHST, CLHST, CLET, and CLLET reached approximately 10^-4 under FUMC and FISC noisy training.FUMC using LHST reached approximately 10^-5.
  • Analytical scope: VQC’s proven resilience differs between FUMC and FISC, with FUMC covering additional Pauli gate and non-unital Pauli noise models.The authors note that whether either approach is intrinsically more resilient remains unsettled, although numerics found no significant difference.
  • Numerical evidence: IBM’s 14-qubit Melbourne noise model produced noiseless costs near 10^-4 for Toffoli and QFT, and near 10^-5 for W-state preparation.The model includes non-unital Pauli noise throughout W = V†U, beyond the paper’s theorem and corollary assumptions.
  • Scope: Theorems and corollaries are restricted to complete ansätze, whereas Fig. 7 also studies incomplete ansätze numerically.The numerics typically obtained the same noiseless cost with noisy and noiseless training.
  • Open questions: The paper identifies future work on more general noise, incomplete ansätze, parameter training, approximate resilience, and resilience for other VHQCAs.It specifically connects VQC to VQE and reports OPR for VQE only for specific Hamiltonian forms.

Appendix A: Preliminaries

The appendix establishes Pauli- and Clifford-based preliminaries and defines noisy entangling, disentangling, and measurement channels used in the resilience proofs.

  • Preliminaries: The proofs use the Pauli product basis, Pauli channels, Clifford unitaries, maximally entangled states, and the all-zero state.These definitions provide the algebraic framework for the appendix’s noise-resilience arguments.
  • Pauli-channel lemma: A Clifford unitary followed by a Pauli channel can be represented using another Pauli channel, as stated in Lemma 1.The proof relies on Clifford normalization of the Pauli group and the channel definition.
  • Noisy entangling gates: The FUMC entangling gate is a tensor product of Hadamard-then-CNOT operations, with Pauli channels placed before and after these gates.This noise model allows correlated Pauli noise around each Hadamard and CNOT.
  • Noisy disentangling gates: The noisy disentangling channel is defined as the adjoint of the noisy entangling channel, with LHST applying it only to the measured qubit pair.Global Pauli channels are still assumed around the relevant Hadamard and CNOT gates.
  • Measurement noise: Measurement noise replaces each ideal zero-state projector with weighted zero and one projectors satisfying p^(j)_00 > p^(j)_01.The resulting noisy POVM element is used in later effective-operator derivations.

1. Effective noisy measurement operator for the HST

For the HST, the appendix derives the effective noisy measurement operator by propagating the noisy all-zero POVM backward through the noisy entangling channel.

  • Operator construction: The HST’s effective noisy POVM is obtained by evolving the noisy all-zero measurement element under the noisy entangling channel in the Heisenberg picture.This accounts for the disentangling unitary preceding measurement in the noiseless circuit.
  • Pauli-basis derivation: The derivation expands computational-basis projectors in the Pauli basis before applying channel linearity and the entangling-channel identities.These steps connect the noisy measurement operator to the appendix’s Pauli-channel preliminaries.

2. Effective noisy measurement operator for the LHST

For the LHST, the appendix derives pairwise noisy measurement operators and combines them into the overall effective noisy POVM.

  • Pairwise operator: Each LHST run measures a pair AjBj after applying the pair’s noisy disentangling channel, producing a pair-specific effective POVM element.The construction evolves Q^(j)_00 under the adjoint noisy disentangling channel.
  • Overall POVM: The overall LHST noisy POVM is formed from the effective operators for the measured pairs and the corresponding channel-dependent coefficients.The appendix defines these coefficients using the Pauli-basis expansion and allows Pauli channels to vary between runs.

Appendix D: Proof of Theorem 1

Appendix D proves that HST-based cost functions retain the same globally optimal unitaries under Noise Model 1. The proof isolates noise-independent terms and shows the noisy objectives are optimized by the noiseless solution set.

  • Setup: The cost evaluation has the form CQC(V) = Tr[ΛEV(ρ)], with EV a V-dependent noisy unital channel.The framework represents the state evolution and measurement through a POVM element Λ.
  • Depolarizing noise: Global depolarizing noise can be added continuously without changing the optimal variational unitaries.Lemma 2 shows strong-OPR persists when global depolarizing channels act throughout the computation.
  • Theorem 1: Theorem 1 establishes strong-OPR for CHST and CLHST under Noise Model 1.The optimal variational unitaries are unchanged by the specified noise model.
  • Proof strategy: The proof decomposes the HST circuit into entangling, V†U, and disentangling intervals, tracking depolarizing channels across each interval.Unitality of the noisy entangling and disentangling channels permits the reduction used in the proof.
  • Optimizer set: Terms independent of W = V†U do not affect global optima, leaving a relevant term whose maximizers coincide with the noiseless objective.The resulting optimizer set includes unitaries satisfying W = eiφ1.

Appendix E: Proof of Theorem 2

Appendix E proves that CHST and CLHST preserve their optimal variational unitaries under the broader Noise Model 2. The argument removes terms independent of W and applies the same optimizer characterization as in Appendix D.

  • Theorem 2: Theorem 2 establishes strong-OPR for CHST and CLHST under Noise Model 2.The proof covers global depolarizing noise together with global Pauli and non-unital Pauli processes specified by the model.
  • CHST: Noise-dependent terms independent of W = V†U do not affect the global optima in the CHST proof.The remaining W-dependent term is optimized by the same unitary set as the noiseless HST cost.
  • CLHST: The CLHST proof similarly separates a W-independent contribution from the term that determines the optimum.The W-independent component g2(V) can be ignored when optimizing the noisy objective.

Appendix F: Proof of Theorem 3

Appendix F proves weaker noise resilience for CLET and CLLET under Noise Model 3. The proof uses rearrangement inequalities and shows that noise restricts the optimizer set to permutations compatible with ordered probability vectors.

  • Theorem 3: Theorem 3 establishes weak-OPR for CLET and CLLET under Noise Model 3.The proof considers Pauli noise at the initial time and measurement noise, while global depolarizing resilience follows from Lemma 2.
  • Objective reduction: The noisy objective reduces to a doubly stochastic mixing of probability vectors associated with the input state and measurement.The matrix elements wil = |⟨i|W|l⟩|² form a doubly stochastic matrix.
  • Rearrangement inequality: The rearrangement inequality bounds the relevant overlap by p↓·q↓, with equality for permutations mapping the ordered bases of p↓ and q↓.The saturating matrices form a subset S of the permutation group.
  • Degeneracy: Equal-magnitude components in p↓ or q↓ make the saturating permutation set S degenerate.Ties in either ordered vector produce multiple equivalent permutations.
  • Optimizer sets: For CLET and CLLET, the noisy optimizer sets are restricted to unitaries whose W = (V′)†U belongs to appropriate permutation subsets.The resulting restricted sets establish weak-OPR rather than strong-OPR for Noise Model 3.

Appendix G: Proof of Corollaries 1-8

Appendix G extends the resilience results to noise occurring during implementation of W = V†U. The corollaries cover Pauli, non-unital Pauli, depolarizing, Clifford, and tensor-product structures under stated conditions.

  • Pauli noise during W: Corollary 1 extends strong-OPR for CHST and CLHST to global Pauli channels acting during implementation of W.The overall noise on system A is required to have the specified composed-channel form.
  • Clifford specialization: Clifford decompositions of W are a special case covered by Corollary 1.Lemma 1 supplies the condition needed for the Clifford specialization.
  • Tensor-product structure: Corollary 3 covers local depolarizing noise when W is tensor-product structured up to an intermediate time.The channels act separately on subsystems A′ and A′′ with the stated tensor-product composition.
  • Non-unital Pauli noise: Corollaries 4 and 5 extend strong-OPR to non-unital Pauli noise during W, including a fixed-time process for CHST with continuous global depolarizing noise.Corollary 4 requires the overall channel to satisfy the stated Pauli-channel condition.
  • Weak-OPR extensions: Corollaries 6–8 extend weak-OPR for CLET and CLLET when interleaved Pauli noise can be represented as a Pauli channel followed by W.The extensions include Clifford and tensor-product cases under their respective composition conditions.
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