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Error mitigation for short-depth quantum circuits

Kristan Temme, Sergey Bravyi, Jay M. Gambetta

arXiv:1612.02058v3quant-phcond-mat.other

TL;DR

The paper addresses estimating observables after noisy quantum evolution and presents two error-mitigation approaches: zero-noise Richardson extrapolation and quasi-probabilistic representation of ideal circuits by noisy ones. In numerical simulations, cancellation reduced the median precision error from approximately 0.15 to 0.05 using the same resources.

  • Problem

    The paper seeks to estimate expectation values of observables after noisy evolution, where sampling introduces additional error into the measured statistic.

  • Method

    The paper applies Richardson extrapolation to reach the zero-noise limit and represents ideal circuits as quasi-probabilistic mixtures of noisy circuits for probabilistic error cancellation.

  • Results

    0.05 median simulation precision δ(β) with error cancellation versus 0.15 median δ0(β) without cancellation, using M = 4,000 runs for each circuit.

  • Takeaways & Limitations

    Error cancellation significantly improves simulation precision in the reported numerical analysis, while quasi-probability overhead increases with the circuit's noisy operations.

  • Takeaways & Limitations

    The analysis assumes specific noise conditions, including time-independent noise in one setting, an initially factorized system-bath state, and an i.i.d. sampling hypothesis.

Abstract

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Two schemes are presented that mitigate the effect of errors and decoherence in short depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and don't require additional qubit resources, so to be as practically relevant in current experiments as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasi-probability distribution.

Reducing noise by Richardson extrapolation

Richardson extrapolation estimates the zero-noise expectation value by expanding noisy evolution in a noise parameter and accelerating convergence across rescaled noise rates. The analysis considers time-independent and more general noise models under stated initial-state and bath assumptions.

  • Reducing noise by Richardson extrapolation: Richardson extrapolation is used to improve convergence toward the zero-noise expectation value in short-depth quantum circuits.The method assumes a noise process constant in time and independent of rescaling the system Hamiltonian parameters.
  • Reducing noise by Richardson extrapolation: The target observable is evaluated after noisy evolution from an initial state, with measurements represented as the noisy expectation plus sampling error.The measured statistic is ˆEK(λ) = EK(λ) + δ, and repeated experiments introduce the sampling error.
  • Reducing noise by Richardson extrapolation: The noise models include time-independent Lindblad dynamics and more general Hamiltonian system-bath interactions, including possibly non-Markovian noise.The latter setting uses an initial product state and an observable supported only on system degrees of freedom.
  • Reducing noise by Richardson extrapolation: The analysis assumes a system initial state that is separable from a bath steady state with respect to the bath Hamiltonian.The observable is restricted to the system through A = AS ⊗1.
  • Reducing noise by Richardson extrapolation: Expectation-value estimation is performed by sampling the noisy evolved state, so finite measurement statistics contribute independently of the evolution noise.The supplied passages state that experiments are typically repeated M ≫1 times under an i.i.d. hypothesis.

I Series expansion in the noise parameter

The noisy expectation value admits a power-series expansion in the noise parameter, with the zeroth-order term equal to the noise-free evolution. Interaction-picture analysis and norm bounds control the remainder and coefficient growth.

  • I Series expansion in the noise parameter: The interaction-picture transformation yields a series expansion of the noisy expectation value in powers of λ.The expansion is recursively obtained from the interaction-picture evolution equation and then converted back to the Schrödinger picture.
  • I Series expansion in the noise parameter: The λ0 contribution equals the noise-free expectation value E∗ sought by extrapolation.The coefficients and remainder are obtained by pairing the expansion terms with the observable trace.
  • I Series expansion in the noise parameter: The remainder is bounded using the Cauchy mean value theorem, Hölder’s inequality, and unitary invariance of Schatten norms.These bounds apply to the interaction-picture noise maps and the observable-state pairing.
  • I Series expansion in the noise parameter: For local noise, the expansion coefficients typically scale as lk = O(N^k) and satisfy |ak| ≤ O((N T)^k).The scaling follows from an extensive local noise norm, ||L||1→1 = O(N).
  • I Series expansion in the noise parameter: If only even powers of λ occur, Richardson extrapolation can achieve higher precision with fewer noise-rate values.The efficiency gain follows from the absence of some powers in the expansion.

II Experimental rescaling of the noise parameter

Because experiments may not directly control λ, the protocol rescales the Hamiltonian evolution and runtime to produce effective evaluations at different noise rates. This enables Richardson extrapolation when the dissipator is constant and independent of Hamiltonian couplings.

  • II Experimental rescaling of the noise parameter: The protocol addresses the inability to directly control λ by rescaling the time-dependent Hamiltonian evolution.The rescaling redefines the coupling parameters and changes the runtime.
  • II Experimental rescaling of the noise parameter: The transformation Jα(t) → c^-1Jα(c^-1t) with runtime cT produces an effective noise-rate change from λ to cλ.The rescaled density matrix is related to the original evolution at the corresponding effective rate.
  • II Experimental rescaling of the noise parameter: This Hamiltonian rescaling allows the experimenter to evaluate EK(λ) at multiple values cλ for Richardson extrapolation.The construction applies to any constant dissipator L that does not depend on the Hamiltonian couplings Jα(t).
  • II Experimental rescaling of the noise parameter: Alternative experimental methods may rescale λ directly, provided the modified noise rates λj are sufficiently accurate for extrapolation.Photon-loss experiments are given as an example where directly changing the loss rate may be preferable.

III Error bounds on the noise-free estimator

The estimator combines measurements from n+1 circuits run at rescaled noise rates, with coefficients chosen to cancel successive powers of λ. Its error bound separates extrapolation remainder from measurement error.

  • III Error bounds on the noise-free estimator: The protocol selects n+1 rescaling parameters c0 = 1 < c1 < . . . < cn and runs the corresponding rescaled evolutions.Each rescaled evolution has runtime Tj = cjT and effective noise rate cjλ.
  • III Error bounds on the noise-free estimator: The coefficients γj are chosen so the weighted measurements cancel the first n powers in the noise expansion.The resulting estimator is constructed from the values ˆEK(cjλ).
  • III Error bounds on the noise-free estimator: The estimator’s error is bounded by combining coefficient-weighted sampling errors with the truncated-series remainder.The bound uses the maximum measurement error δ∗ = maxj |δj| and the remainder bound at each rescaled rate.
  • III Error bounds on the noise-free estimator: Bulirsch–Stoer and harmonic sequences provide alternative choices for the rescaling parameters cj.The supplied passage describes exponential and harmonic series used in Richardson extrapolation.

Probabilistic error cancellation by resampling

Probabilistic error cancellation represents an ideal circuit as a quasi-probabilistic mixture of noisy circuits and estimates ideal expectation values by sampling those circuits. The construction requires suitable noisy-operation bases and can preserve qubit count, circuit length, and depth, but may require detailed tomography and has practical overhead limitations.

  • The scheme represents an ideal circuit as a quasi-probabilistic mixture of noisy circuits.The noisy circuits are sampled according to a probability distribution with efficiently computable signed coefficients.
  • Applying the method requires sufficiently accurate gate tomography and efficiently computable sampling coefficients.The noisy operations must be characterized with accuracy comparable to the desired accuracy of the ideal circuit.
  • A noisy-operation basis must be sufficiently large to express any ideal unitary gate as a linear combination of noisy operations.The operations are assumed to form a full basis of trace-preserving completely positive operations.
  • Ideal expectation values are obtained by estimating expectation values from a suitable random ensemble of noisy circuits.The noisy and ideal circuits can use the same number of qubits and the same length.
  • For general operation-dependent noise, constructing QPRs with optimal overhead remains an open problem, and one stated upper-bound coefficient lacks practical implications.A preliminary Clifford+T analysis gives γβ ≤ 1 + O(ϵ) when noisy operations are ϵ-close to ideal ones, but the constant is too large for practical use.
  • The method can preserve circuit depth: noisy circuits corresponding to an ideal circuit of depth d have depth at most d.The construction may nevertheless require intermediate qubit initializations for some noise models, including amplitude damping.

IV Minimal overhead decomposition of noise free circuit

The product-QPR construction decomposes each ideal gate into a minimal-overhead quasi-probability representation and combines these gate-wise representations across the circuit. Locality restricts the required noisy operations to those acting within each gate’s support, including at most two qubits.

  • For each ideal Clifford+T gate, a linear program over noisy operations finds a representation minimizing the sum of absolute coefficients.The candidate operations are those whose support is contained in the support of the ideal gate.
  • The optimal solution satisfies µα = |ηα|, and the resulting γβ defines the gate’s QPR overhead.This equality follows because otherwise the linear-program objective could be decreased.
  • A circuit-level product QPR multiplies gate-wise overheads and combines their probability and sign distributions across gates.The circuit distributions factor as products of the individual gate distributions and signs.
  • Because noisy operations act only within each ideal gate’s support, the decomposition can be restricted to operations acting on at most two qubits.Such operations can therefore be represented using local noisy bases for the one- and two-qubit gates.

V Depolarizing noise cancellation and numerical results

For depolarizing noise, the method constructs noisy bases by appending Pauli maps to ideal gates and samples modified circuits with gate-specific probabilities. The resulting quasi-probability representation gives an explicit single-qubit overhead and analogous two-qubit cancellation through Pauli maps.

  • The depolarizing noisy basis consists of operations Oα = DkPU, combining a k-qubit depolarizing channel, Pauli map, and ideal gate.Here Dk returns the maximally mixed state with probability ϵ and otherwise does nothing.
  • For a single-qubit gate, the optimal coefficients are η1 = 1 + 3ϵ/[4(1 − ϵ)] and ηα = −ϵ/[4(1 − ϵ)] for the other three Pauli maps.These coefficients minimize the sum of absolute coefficients in the single-qubit decomposition.
  • The single-qubit overhead is γβ = (1 + ϵ/2)/(1 − ϵ).The CNOT is treated similarly by representing the inverse two-qubit depolarizing channel as a linear combination of two-qubit Pauli maps.
  • Randomized circuits are generated by adding single-qubit Pauli errors with p1 = ϵ/(4 + 2ϵ) and two-qubit Pauli errors after CNOTs with p2 = ϵ/(16 + 14ϵ).Unmodified gates occur with probabilities 1 − 3p1 for single-qubit gates and 1 − 15p2 for CNOTs.
  • The final readout strings from the noisy circuit samples are used to estimate E*(β), with the sign determined by the number of inserted Pauli operators.The sign is σβ(α) = (−1)^r when r Pauli operators were added.

Numerical simulations

Numerical simulations compare probabilistic error cancellation with direct noisy execution on randomly generated Clifford+T circuits. With identical run budgets, cancellation substantially improves the estimated output-probability precision.

  • Simulation setup: 500 random Clifford+T circuits were simulated with n = 6 qubits, depth d = 20, error rate ϵ = 0.01, and M = 4,000 total runs.The circuits used alternating single-qubit-gate and CNOT layers, with initial state |+⟩^⊗n.
  • Estimation procedure: The simulations estimate E∗(β) by sampling noisy circuits from a randomized ensemble and allocating the total run budget across circuit groups.The estimator is unbiased for any allocation, while the chosen allocation minimizes its variance for fixed M.
  • Results: Error cancellation achieved median simulation precision δ(β) approximately 0.05 across the random circuits.Here δ(β) = |Ê(β) − E∗(β)|, where Ê(β) is the error-cancelled estimate.
  • Results: Direct execution without error cancellation yielded median simulation precision δ0(β) approximately 0.15 using the same M = 4,000 runs.The authors conclude that error cancellation significantly improves simulation precision.

VI Quasi-probability representation for amplitude-damping noise

The amplitude-damping construction extends quasi-probability error cancellation with noisy state preparations because noisy damped gates alone cannot represent arbitrary ideal unitaries. The resulting representation preserves circuit depth while assigning an overhead to each gate type.

  • Motivation: Amplitude-damping noisy unitary gates A^⊗kU alone cannot simulate arbitrary ideal unitary gates because the damping channel is non-unital.The construction therefore extends the noisy basis with state-preparation maps and uses non-product quasi-probability representations.
  • Noisy basis: The extended noisy basis includes noisy preparations of |+⟩, |−⟩, |0⟩, and |1⟩, noisy single-qubit Clifford+T gates, and noisy two-qubit CNOT-related operations.The basis is claimed to simulate any ideal Clifford+T circuit.
  • Circuit representation: The quasi-probability representation preserves the ideal circuit depth, although it does not have a simple product form.This construction applies to any ideal Clifford+T circuit in the stated noisy basis.
  • Overhead construction: The ideal |+⟩ state preparation is represented with overhead γ′ = Σα|ηα| ≤ γ, where γ is the single-qubit overhead defined earlier.The resulting circuit construction replaces CNOTs first, then replaces remaining single-qubit gates and |+⟩ preparations by their quasi-probability representations.
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