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Practical Quantum Error Mitigation for Near-Future Applications

Suguru Endo, Simon C. Benjamin, Ying Li

arXiv:1712.09271v2quant-ph

TL;DR

Near-future quantum devices need practical ways to reduce errors without the resource cost of full fault tolerance. The paper uses gate set tomography and circuit-based QEM constructions, including optimized quasi-probability and exponential extrapolation methods. It reports accurate mitigation for localized Markovian errors in simulations up to 19 qubits, while noting longer computation time as the cost of systematic error negation.

  • Problem

    Full fault tolerance carries an enormous resource cost, while existing QEM techniques assume full knowledge of the error model and lack explicit circuit-derivation procedures.

  • Method

    The paper measures noise with gate set tomography, derives QEM circuits using single-qubit Clifford gates and measurements, and compares quasi-probability decomposition with exponential error extrapolation.

  • Results

    The protocol fully eliminates localized Markovian-error effects in expected-value estimation, and exponential extrapolation is reported as accurate in simulations of up to 19 qubits.

  • Takeaways & Limitations

    QEM can enhance quantum-computer performance at small-to-medium scale where full code-based error correction is impossible, within the studied scope.

  • Takeaways & Limitations

    Systematic error negation requires quantum computation to run for longer than an error-free system.

Abstract

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It is vital to minimise the impact of errors for near-future quantum devices that will lack the resources for full fault tolerance. Two quantum error mitigation (QEM) techniques have been introduced recently, namely error extrapolation [Li2017,Temme2017] and quasi-probability decomposition [Temme2017]. To enable practical implementation of these ideas, here we account for the inevitable imperfections in the experimentalist's knowledge of the error model itself. We describe a protocol for systematically measuring the effect of errors so as to design efficient QEM circuits. We find that the effect of localised Markovian errors can be fully eliminated by inserting or replacing some gates with certain single-qubit Clifford gates and measurements. Finally, having introduced an exponential variant of the extrapolation method we contrast the QEM techniques using exact numerical simulation of up to 19 qubits in the context of a `SWAP test' circuit. Our optimised methods dramatically reduce the circuit's output error without increasing the qubit count or time requirements.

I. INTRODUCTION

Near-term quantum devices need error mitigation because full fault tolerance requires enormous resources, while QEM targets error-free observable estimates from imperfect circuits. This paper addresses imperfect error-model knowledge, derives practical mitigation circuits, and evaluates optimized methods numerically.

  • Motivation: Full quantum fault tolerance can detect and correct physical-qubit errors using logical qubits, but requires an enormous multiplicative resource cost.This motivates mitigation approaches for near-term devices and shallow hybrid algorithms.
  • Existing QEM techniques: Error extrapolation estimates the zero-error observable by comparing results at minimum and deliberately increased error rates.The approach presumes that error sources scale proportionately.
  • Existing QEM techniques: Quasi-probability decomposition samples modified, error-burdened circuits and assigns parity-dependent weights to obtain an unbiased estimator.The sampled circuit distribution depends on the noise model.
  • Paper contributions: The paper resolves practical gaps by using gate set tomography to characterize errors and by specifying how to derive extrapolation and quasi-probability circuits.The authors state that preparation and measurement noise in gate set tomography is not harmful to the overall QEM approach.
  • Paper contributions: Single-qubit Clifford gates and measurements can form a complete set of operations for compensating localized Markovian errors in expected-value estimation.The paper focuses on estimating observables from final quantum-circuit states, including SWAP-test outputs.
  • Evaluation: An exponential error-extrapolation model is introduced alongside optimized quasi-probability and extrapolation methods, then tested on SWAP-test circuits of up to 19 qubits.The simulations include 15 qubits across a comprehensive set of cases and 19 qubits for specific cases.

V. PER-OPERATION ERROR CORRECTION

The quasi-probability method represents error correction through sampled initial states and per-operation replacements. Circuit outcomes are combined using probability and sign factors associated with those replacements.

  • Initial-state correction: The method assumes a set of initial states whose ideal state can be represented through weighted sampled states.The supplied passage introduces the initial-state decomposition but does not provide the complete expression.
  • Per-operation correction: Each ideal operation is replaced by a sampled error-burdened operation selected according to an associated probability.The construction applies operation by operation within the circuit.
  • Estimator construction: The mitigated estimate combines sampled-circuit outcomes with effective signs determined by the sampled input, operations, and output.The passages identify sign factors and the effective outcome used in estimating the ideal observable.

VI. VARIANCE AMPLIFICATION IN QUASI-PROBABILITY DECOMPOSITION

Quasi-probability error mitigation reaches the ideal observable estimate by sampling modified circuits, but negative quasi-probabilities amplify variance and increase the required number of samples. The overhead grows with the mitigation cost of individual operations and with circuit size.

  • Negative quasi-probabilities amplify the variance of the estimated observable.
  • The error-mitigated computation requires more samples than error-free computation to achieve the same accuracy.The supplied passages express this comparison through the sample-ratio relation involving C and ⟨Q(0)⟩.
  • The required sample count scales as Nr ∼(C/ϵ)^2 when the standard deviation is limited to σ ∼ϵ.A larger mitigation factor C therefore increases computing time.
  • Because C is the product of per-operation factors, the overall mitigation cost increases with the number of operations.Fully connected quantum computers can reduce communication operations and may thereby reduce this cost.
  • A universal expected-value operation set uses measurement and single-qubit Clifford gates, with sixteen linearly independent basis operations spanning any single-qubit operation.Multi-qubit operations use tensor products of these basis operations.
  • The basis-operation set can represent non-Clifford gates, but direct synthesis may incur an unacceptably steep exponential time overhead.The practical protocol instead assumes direct implementation of a universal gate set and compensates for slight imperfections.

VIII. ERROR MITIGATION USING BASIS OPERATIONS

Basis operations provide two ways to mitigate localized errors: compensate the error component or invert the noise after an imperfect operation. Tensor-product decompositions extend the approach to multi-qubit operations, while error boosting supports extrapolation.

  • An operation with error can be corrected using sixteen basis operations to realize the corresponding error-free operation.The paper presents compensation and inverse methods for this decomposition.
  • Compensation method: The compensation method retains the original operation and decomposes only its error component using basis operations.Its coefficients are uniquely determined when the basis operations are linearly independent.
  • Inverse method: The inverse method applies the inverse noise after an imperfect operation to realize the error-free operation.This method requires the imperfect operation matrix O to be invertible.
  • Error-mitigation circuits randomly choose between the original gate and basis operations, while extrapolation schematically fits error-dependent outputs.The circuit choice is determined by random numbers, and identity gates require mitigation unless memory error is negligible.
  • Multi-qubit errors, including errors that entangle qubits, can be mitigated using tensor products of non-entangling basis operations.The controlled-NOT example supports the claim that errors in that gate form can be mitigated using basis operations.
  • Initialisation and measurement errors can also be corrected by decomposing transformations that map noisy states or observables to their error-free forms.
  • The same decomposition technique can increase an operation's error for alternative error extrapolation.Error boosting can avoid negative quasi-probabilities in examples such as boosting Pauli errors with Pauli gates.

IX. QUANTUM GATE SET TOMOGRAPHY

Gate-set tomography (GST) estimations can be used in quantum error mitigation despite imperfect knowledge of the physical system, because a shared similarity transformation preserves expected values. The mitigation cost depends on the error model and can be reduced by choosing the correction method and using Pauli twirling under suitable conditions.

  • GST-based error mitigation: GST estimations may be imperfect, yet their shared similarity transformation preserves the circuit’s expected observable value exactly.The transformation applies consistently to operations, initial states, and observables, so it does not introduce computing error.
  • GST-based error mitigation: The physical system can therefore produce the desired error-free output using decompositions based on GST-estimated states, observables, and operations.The cost is a potential increase in the number of samples required.
  • Cost estimation: For depolarising noise, the inverse method has lower correction cost than the compensation method, whereas Pauli twirling lowers cost for over-rotation noise.The comparison uses the universal operation set and includes initialisation, measurement, memory, single-qubit, and two-qubit errors.
  • Cost estimation: 16^n parameters must be optimised for an n-qubit quantum computer, making GST-based cost optimisation non-trivial.Under reasonable conditions, Pauli twirling can reduce the optimisation cost.
  • Cost estimation: C −1 ≃ aϵ for a two-qubit entangling gate under depolarising noise, with a between 2 and 3 and approaching 2 under favourable conditions.Across the shown noise range, depolarising noise generally gives an approximate upper bound on mitigation cost, although lower-fidelity gates can exceed it.
  • Cost estimation: (1 + 2ϵ)2N ≈ e4Nϵ repetitions amplify the standard-deviation cost across N corrected gates; Nϵ = 2 implies approximately 3,000 repetitions.Larger overhead factors may still be acceptable depending on quantum-computer speed.

XI. NUMERICAL SIMULATION

The simulations evaluate QEM on noisy SWAP-test circuits, comparing quasi-probability decomposition with linear and exponential extrapolation under Pauli and leakage errors. Exponential extrapolation most closely approaches the ideal result while preserving qubit count and total experimental runs.

  • Simulation setup: The error models apply local noise around initialization, measurement, and gates, with total rates of 0.08% for initialization and measurement, 0.16% for single-qubit gates, and 0.32% for two-qubit gates.The models include inhomogeneous Pauli and leakage errors, with the same noise applied at several circuit locations.
  • QEM comparison: With 10^4 runs, both QEM methods shift estimates closer to the ideal ⟨Z⟩=0.5 than no mitigation; quasi-probability removes systematic bias but produces wider distributions.For inhomogeneous Pauli errors, the three means are 0.1961, 0.3415, and 0.5011; for leakage errors, they are 0.3819, 0.4710, and 0.5007.
  • Simulation setup: The SWAP-test simulations use 19 qubits for inhomogeneous Pauli errors and 15 qubits for leakage errors, with leakage requiring an additional qubit.The probe qubit is accompanied by two groups of qubits; the leakage model uses fewer circuit qubits because one additional simulation qubit models leakage.
  • QEM comparison: Exponential extrapolation outperforms linear extrapolation for both noise types and approaches the ideal value more closely.The method assumes that the observable decays exponentially with error rate and uses two data points under that assumption.
  • QEM comparison: The exponential method yields mean estimates of 0.5111 for inhomogeneous Pauli errors and 0.4986 for leakage errors.Expected absolute errors are 0.06501 and 0.01882, respectively; the leakage result is within a factor of three of the ideal shot-noise limit 0.00691.
  • Scaling: For fully parallel circuits, today’s error rates are insufficient much beyond 50 qubits, whereas tenfold lower rates suffice for 80 qubits and beyond.The SWAP-test circuit is feasible in the supremacy regime with today’s best fidelities, while full parallelism is more demanding.

XII. INTUITION FOR EXPONENTIAL EXTRAPOLATION

The paper explains exponential extrapolation by modeling the accumulated effect of many local noise events. It also concludes that systematic QEM can remove bias for localized Markovian noise, though it requires longer computation than an error-free system.

  • Noise model: The circuit’s total noise is modeled as the accumulation of N noise-burdened operations, with each error component only weakly dependent on the error rate.The argument temporarily ignores computing operations because they do not affect the general noise-counting explanation.
  • Noise model: The number of error events follows a binomial distribution that can be approximated by a Poisson distribution.This approximation produces the exponential dependence used by the extrapolation method.
  • Exponential dependence: The observable’s noise impact is proportional to e^(-Nϵ), explaining why exponential extrapolation can outperform linear extrapolation.The exponential form arises from the accumulated probability of noise events across the circuit.
  • Protocol scope: Following the protocol, an experimentalist can construct a QEM algorithm with zero bias without prior knowledge of the physical noise, provided the noise is localized and Markovian.The protocol uses measured noise effects to design compensating circuits.
  • Protocol scope: Single-qubit Clifford gates and measurements can provide a complete set of operations that compensate arbitrary localized Markovian noise.Gate-set tomography is presented as a tool for measuring the noise relevant to compensation.
  • Trade-off: The systematic method requires longer quantum computation, while exponential extrapolation can provide accurate estimates with shorter computing time.The paper suggests combining the methods to optimize accuracy and efficiency.

Appendix A: Pauli transfer matrix

The appendix represents quantum states, observables, and operations as real vectors and matrices in the Pauli basis. It then uses matrix invertibility to construct the decompositions required for error mitigation.

  • Representation: Pauli operators form a real-vector representation of n-qubit states and observables, with Hilbert-space dimension d=2^n.States are represented as real column vectors, while observables are represented as real row vectors.
  • Representation: A physical operation is represented by a real square matrix acting on the state vector in the Pauli-transfer representation.If ρ′=O(ρ), then the corresponding vectors satisfy |ρ′⟩⟩=O|ρ⟩⟩.
  • Operation decomposition: The controlled-NOT gate is decomposed into basis operations, and the corresponding quasi-probability cost is C=9.The decomposition expresses the gate as a signed combination of operations used by the mitigation procedure.
  • Invertibility: If the noisy basis-operation matrix remains sufficiently close to the noiseless matrix, it stays invertible and supports a valid decomposition.The appendix uses singular-value bounds to establish invertibility under bounded error.
  • Operation decomposition: For n-qubit operations, the decomposition coefficients are obtained by applying the inverse tensor-product matrix (A1⊗···⊗An)^-1 to the operation vector.Each qubit contributes a set of 16 basis operations ordered by the Pauli operators I, X, Y, and Z.
  • State and observable decomposition: Four linearly independent prepared states and four linearly independent measured observables allow decomposition of the ideal input state and observable under noisy operations.Clifford gates generate the required state-preparation and measurement bases; invertibility of the corresponding matrices determines the coefficients.

Appendix E: Quantum gate set tomography

The appendix develops GST-based estimates of noisy operations, states, and observables, showing that transformed estimates still predict observable expectations correctly. It also gives invertibility conditions for the reconstructed gate representation.

  • Error calibration: Initialization and measurement errors are incorporated into separate input and output matrices rather than measured independently.Taking the operation to be identity enables these calibration matrices to be measured.
  • Representation freedom: The reconstructed operation is related to the physical operation by an invertible transformation shared across operations.The matrix relation is expressed as ˆO = Tg^-1 ˜OT^-1.
  • Prediction invariance: GST estimates may differ from the physical states, observables, and operations yet still predict the correct expected value for an observable after any operation sequence.The estimated and physical representations are related by a common transformation, preserving the sequence prediction.
  • Stability condition: The reconstructed matrices remain invertible when initialization and output error severities are below the corresponding minimum singular values.The stated conditions are ¯εin < smin(Min(0)) and ¯εout < smin(Mout(0)); the chosen values are approximately 0.3311 and 1.

Appendix G: Upper bound of the cost

This appendix bounds the quasi-probability cost of compensating errors in operations, observables, and initial states. The bounds depend on decomposition size, error severity, and basis stability.

  • Operation compensation: Operation-error compensation represents the error E = O(0) − O in a basis-operation decomposition, with cost determined by the resulting coefficients.The compensation method takes λ = 1 and realizes the ideal operation as O(0) = O + E.
  • Operation compensation: The quasi-probability decomposition coefficients are obtained by solving q = (A1 ⊗ · · ·⊗An)^−1E over 16^n-dimensional vectors.The basis matrix is 16^n-dimensional, so the construction contains 16^n decomposition coefficients.
  • Stability assumption: The coefficient bound assumes ϵmax < smin(A(0))/16, linking correction stability to the minimum singular value of the ideal basis matrix.Here ϵmax is the largest basis-operation deviation from its ideal value.
  • Initialization compensation: Initial-state correction uses ϵin = ∥Min − Min(0)∥max and bounds the cost through the inverse factor [smin(Min(0)) − ϵin]−1.The bound relies on ∥Min−1∥max ≤ 4[smin(Min(0)) − ϵin]−1.
  • Observable compensation: Observable-error correction uses ϵout = ∥Mout − Mout(0)∥max as its error-severity measure.The corresponding correction cost is derived from this observable error measure.

Appendix H: Error models

The error-model appendix specifies the gate, preparation, measurement, memory, and noise models used in the simulations. It distinguishes operation timing and measurement outcomes while including several noise families.

  • Gate set: The simulated gate set includes initialization, projective measurement, single-qubit Clifford gates, a non-Clifford T gate, and a two-qubit maximally entangling gate.The two-qubit gate is equivalent to controlled-NOT and controlled-phase gates up to single-qubit gates.
  • Measurement model: Measurement errors can merge several POVM outcomes into the two reported outcomes ν = 0, 1.This models imperfect access to the underlying measurement outcome k.
  • Gate noise: Noisy gates are modeled as G = NaG(0)Nb, allowing any noisy gate to be represented around its ideal operation.Because G(0) is invertible, one factor can always be chosen as the identity.
  • Timing and memory: A computing cycle assigns the measurement and two-qubit-gate time cost, with at most one operation performed on each qubit per cycle.Single-qubit gates occur midway through a cycle and idle qubits undergo memory noise.
  • Timing and memory: The model separates identity operations, which take no time, from memory operations, during which waiting qubits can accumulate memory errors.Identity operations are used to measure the calibration matrix g.
  • Noise families: The appendix models single- and two-qubit depolarizing, dephasing, damping, random Hamiltonian, and gate-dependent noise processes.The gate-dependent random-Hamiltonian model uses time-independent noise for repeated implementations of the same gate.

2. Dephasing Error

This section describes dephasing and related gate-dependent error models, then gives the GST procedure used to characterize gates for QEM. The procedure constructs measured operation matrices from noisy preparations and observables.

  • Dephasing error models: Gate-dependent noise is parameterized by an error intensity ϵ and can be specified separately for each operation.The models include explicit noisy single-qubit and two-qubit gates as well as random Hamiltonian noise.
  • Dephasing error models: Random Hamiltonian noise is time independent, so the same gate receives the same noise model whenever implemented.The Hamiltonian is generated from random matrix elements and defines E(1,2)(ϵ) = [e−iϵπH].
  • χ-matrix construction: A noisy operation is generated from its ideal χ-matrix by perturbing it, enforcing trace preservation, and shifting negative eigenvalues when necessary.The final normalization keeps the operation trace preserving and its maximum eigenvalue below 1.
  • GST implementation: Single-qubit GST prepares |0⟩, |1⟩, |+⟩, and |y+⟩, applies the target gate, and measures four observables to construct a 4 × 4 matrix ˜O.The four observables are 11, σx, σy, and σz, and both preparation and measurement may be noisy.
  • GST optimization: The transformation matrix T can vary between qubits but must remain fixed across all gates on a given qubit, and it may be chosen to reduce QEM cost.This freedom provides an optimization parameter for the characterization procedure.
  • GST implementation: Two-qubit GST uses 16 tensor-product initial states and 16 tensor-product observables, producing tensor-product calibration matrices.GST is implemented for each qubit pair on which a two-qubit gate may act.

2. Quasi-probability decomposition

The protocol uses GST estimates to construct quasi-probability decompositions for state preparation, measurements, and noisy gates, then samples modified circuits to implement error mitigation. It accounts for imperfect basis-operation estimates because the final result is invariant under the resulting similarity transformation.

  • GST estimates initial states, observables, gates, and basis operations used to compute the quasi-probability decomposition.These estimates provide the experimentally informed ingredients for the decomposition.
  • For each gate, the inverse method computes the ideal Pauli transfer matrix, inverts the noise, and solves for quasi-probabilities.Single-qubit decompositions are computed separately for each qubit and gate.
  • Two-qubit gates are decomposed using tensor products of single-qubit basis operations to represent the inverse noise.The same procedure is applied with tensor-product basis operations labeled by qubit.
  • The protocol also solves equations for quasi-probabilities that mitigate errors in initial states and measured observables.State-preparation and measurement quantities are treated separately from gate decompositions.
  • Basis-operation estimates can differ from actual operations without changing the final computing result because of similarity-transformation invariance.The decomposition uses estimated basis operations despite this experimental mismatch.
  • Implementation samples random basis, gate, and measurement indices, runs the corresponding modified circuit once, and records the measurement outcome.The circuit initializes selected states, applies noisy gates and basis operations sequentially, and measures the selected observable.
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