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Error-mitigated digital quantum simulation

Sam McArdle, Xiao Yuan, Simon Benjamin

arXiv:1807.02467v5quant-ph

TL;DR

Hardware errors limit variational quantum simulations, motivating efficient error mitigation. The paper proposes a stabiliser-like parity-check method and reports detection of substantial depolarising errors, while identifying scope limits for superposition states and realistic hardware noise.

  • Problem

    Variational quantum simulations are limited by hardware errors, creating a need for efficient error-mitigation methods.

  • Method

    The method checks conserved particle-number parity, requiring no additional ancilla qubits or circuit repetitions beyond the quantum gradient-finding circuit.

  • Results

    80% of single-gate errors can be detected in the M → ∞, M >> N limit, while sequential total, spin-up, and spin-down parity checks detect around 66% of errors.

  • Takeaways & Limitations

    Parity-based detection can benefit calculations under stochastic and correlated noise, including in combination with existing error-mitigation techniques.

  • Takeaways & Limitations

    The 80% limit is unattainable in practice, and parity checks can miss errors in superpositions of multiple Slater determinants.

Abstract

from arXiv · show

Variational algorithms may enable classically intractable simulations on near-future quantum computers. However, their potential is limited by hardware errors. It is therefore crucial to develop efficient ways to mitigate these errors. Here, we propose a stabiliser-like method which enables the detection of up to 60 - 80 % of depolarising errors. Our method is suitable for near-term quantum hardware. Simulations show that our method can significantly benefit calculations subject to both stochastic and correlated noise, especially when combined with existing error mitigation techniques.

SUPPLEMENTARY MATERIALS

The paper formulates molecular simulations using second quantisation, maps fermionic Hamiltonians to qubits with Jordan–Wigner encoding, and exploits particle-number parity conservation.

  • DIGITAL QUANTUM SIMULATION: VQE is used to find ground-state energies of physical Hamiltonians, focusing primarily on molecular simulations in the second-quantised formalism.The procedure is also described as similar for lattice models such as the Fermi-Hubbard model.
  • SUPPLEMENTARY MATERIALS: The fermionic Hamiltonian is constructed from creation and annihilation operators obeying anticommutation relations that enforce wavefunction antisymmetry.The electronic Hamiltonian is written in the second-quantised representation after projection onto finite basis wavefunctions approximating spin-orbitals.
  • DIGITAL QUANTUM SIMULATION: Jordan–Wigner encoding stores each orbital’s occupation number in a qubit state, with |0⟩ unoccupied and |1⟩ occupied.The work uses Jordan–Wigner rather than Bravyi–Kitaev encoding because prior work found Bravyi–Kitaev states more susceptible to errors.
  • DIGITAL QUANTUM SIMULATION: The Jordan–Wigner mapping uses X and Y operators to change orbital occupations, while Z strings enforce electron-exchange antisymmetry.The mapped molecular Hamiltonian is expressed as a linear combination of products of Pauli operators.
  • DIGITAL QUANTUM SIMULATION: Each qubit-Hamiltonian term contains an even number of X or Y operators and therefore conserves particle-number parity.This follows from the even number of creation and annihilation operators in each fermionic Hamiltonian term.

NOISE MODELS

The simulations model stochastic hardware noise with symmetric depolarising channels applied to single- and two-qubit gates, parameterised by gate-malfunction probabilities.

  • NOISE MODELS: A symmetric depolarising channel is applied to all quantum gates in the stochastic error model.The single-qubit gate channel is specified separately from the two-qubit gate channel.
  • NOISE MODELS: The single-qubit channel is parameterised by p, the probability that the gate malfunctions.The two-qubit gate depolarising channel is defined separately.
  • NOISE MODELS: For the two-qubit channel, error operators act on the participating qubits and range over I, X, Y, and Z, excluding the I,I no-error case from the error sum.The no-error contribution is represented by the term outside the sum.

Temporally correlated over/under rotations

The correlated-noise model replaces circuit gates with parametrised equivalents and applies temporally correlated over- or under-rotations through qubit-dependent parameters.

  • Temporally correlated over/under rotations: Temporally correlated over/under rotations are modelled by replacing each discrete circuit gate with a parametrised equivalent.The replacements include Hadamard and CNOT gates.
  • Temporally correlated over/under rotations: All rotation gates use R(εθ), except rotations used to change measurement bases for Hamiltonian-term measurements.Setting ε=1 recovers the original gates.
  • Temporally correlated over/under rotations: For each circuit iteration, each qubit receives a random δq uniformly distributed between ±0.01, producing 0.99 < εq < 1.01.Single-qubit gates use εq = 1 − δq/10, while target qubits of two-qubit gates use εq = 1 − δq.

ERROR DETECTION RATES

Analytic error-detection rates rely on symmetric depolarising, low-error, number- and spin-conserving assumptions, with negligible single-qubit errors relative to two-qubit errors.

  • ERROR DETECTION RATES: The error-detection analysis begins from explicitly stated assumptions about the noise and ansatz circuit.These assumptions are introduced before deriving detection rates.
  • ERROR DETECTION RATES: The derivation assumes symmetric depolarising errors, a low error rate causing at most one malfunctioning gate, and ansatz gates conserving particle number and spin.These assumptions define the setting for the analytic detection-rate results.
  • ERROR DETECTION RATES: Single-qubit gate error rates are assumed negligible compared with two-qubit gate error rates.This assumption simplifies the analytic error-detection-rate derivation.
  • ERROR DETECTION RATES: The method remains applicable under higher noise rates, different noise models, and other number-conserving ansatze, but analytic detection bounds become more difficult.The stated applicability exceeds the conditions used for the analytic bound.

Total electron number parity check

A single total-electron-number parity check detects errors that change electron-number parity, including errors during both ansatz execution and parity-check operations. Under the stated depolarising assumptions, this detects about 53% of errors, while detection during the check sequence can reach about 66% with additional spin checks.

  • Total electron number parity check: 53% of errors are detected by a single total-electron-number parity check under the stated assumptions.The result is also expressed as 8/15 of the 15 possible two-qubit errors.
  • Total electron number parity check: For two spin-orbitals, the method detects 100% of double bit-flip errors, decreasing to 66% for four spin-orbitals.
  • Total electron number parity check: The detectable errors are those that change the electron-number parity of the state vector.
  • Total electron number parity check: The same 8/15 detection rate applies when errors occur during either the ansatz circuit or a non-local parity-check gate sequence.For ansatz errors, the detectable set includes eight specified two-register-qubit errors; for parity-check errors, it includes eight register–ancilla errors.

Spin-up and spin-down parity check

Spin-up and spin-down parity checks extend total-number parity detection to errors that change either spin parity, but they do not detect every two-qubit bit-flip error.

  • Spin-up and spin-down parity check: Spin-up and spin-down parity checks detect additional errors beyond those detected by total-electron-number parity.
  • Spin-up and spin-down parity check: These checks detect certain two-qubit errors that change the value of either spin parity.
  • Spin-up and spin-down parity check: All two-qubit bit-flip errors remain undetectable by the combined parity checks.Errors occurring on two spin-up orbitals or two spin-down orbitals cannot be detected, whereas errors affecting one orbital of each spin can be detected.

Electron number check

The electron-number check measures number bits iteratively, using ancilla-based circuits to identify states with inconsistent electron numbers. Its maximum detection rate is 80%, but this bound depends on restrictive assumptions and can overestimate performance for superpositions of Slater determinants.

  • Electron number check: The circuit measures the first electron-number bit N1 with an ancilla measured in the X basis.The first-bit circuit uses R1 gates given by diag(1, eπi).
  • Electron number check: The second bit is measured iteratively using a phase correction conditioned on the previously measured first bit.For N1 = 1, the correction is ω2 = diag(1, e−πi/2), and measuring the second bit as 1 establishes an electron number of 3 in the example.
  • Detection performance: Under the stated assumptions, parity measurements initially detect 66% of errors, while the idealized maximum detection rate reaches 80% for single-gate depolarising errors.The 80% limit corresponds to detecting 12/15 errors when M →∞ and M >> N, although the passage says this is unattainable in practice.
  • Limitations: Some two-qubit bit-flip errors acting on one occupied and one unoccupied same-spin orbital remain undetectable because they mimic spin-conserving excitations.This creates a concrete class of errors that particle-number and spin-parity checks cannot identify.
  • Limitations: The maximum detection estimate assumes a single Slater determinant, whereas realistic ansatz outputs may be superpositions of multiple determinants.Double bit-flip errors can preserve the checked parities for some determinants while changing particle number for others, making detection probabilistic.
  • Error filtering: When the second-bit measurement yields an inconsistent value, the procedure identifies an error and discards the result.For a superposed state affected by a double bit flip, the second-bit measurement can detect the error with 50% probability.

Non-violation of the variational principle

Filtering removes selected erroneous states while retaining a density matrix over accepted states, so the measured energy remains bounded below by the ground-state energy. The filtered circuit also reproduces the conventional VQE result and can reduce readout noise through single-qubit measurements.

  • Non-violation of the variational principle: The filtering procedure removes states with incorrect particle-number parities from the noisy density matrix.The physical ansatz states occupy a restricted region of Hilbert space, while errors can move states outside that region.
  • Non-violation of the variational principle: Because every normalised state has energy at least Eg, the filtered mixture cannot produce an energy below the true ground-state energy.Equality holds only when the filtering procedure is 100% effective and the system is in the ground state.
  • Measurement consequences: The filtered circuit obtains the same result as the conventional VQE circuit.It does so by measuring a single qubit each time, which the passage states reduces readout noise relative to direct VQE measurement.

ALTERNATIVE STABILISER-VQE CIRCUIT

The alternative circuit measures a Hamiltonian term with an ancilla and then checks conserved register parity to detect errors. It can also support derivative-based variational algorithms, while its readout-error trade-offs and parity coverage remain limited.

  • Alternative measurement circuit: The ansatz prepares a physical state, and a controlled Hamiltonian term h_j enables measurement of ⟨ψ|h_j|ψ⟩.Because h_j is a product of Pauli terms on different qubits, the controlled operation can be decomposed into controlled Pauli gates.
  • Parity-based detection: After ancilla measurement, a parity measurement on the register detects whether the state remains in the physical parity sector.For Jordan–Wigner Hamiltonians, the relevant Pauli terms conserve electron-number parity, so the error-free register is a parity eigenstate.
  • Readout-error behavior: For a single readout error, the alternative circuit usually rejects corrupted register outcomes, whereas ancilla errors can remain undetected.With M qubits, ancilla errors occur with probability 1/M, while register readout errors occur in the other (M − 1)/M cases.
  • Readout-error behavior: The circuit variant checks total electron parity but not spin-up or spin-down parities, and gate errors may dominate because circuits contain many more two-qubit gates than measured qubits.This limits the set of conserved quantities checked by the alternative construction and affects its practical error profile.
  • Parity-based detection: The parity check is effectively free for gradient finding because it requires no additional ancillas or circuit repetitions beyond the gradient procedure.The same construction applies to real-time and imaginary-time variational algorithms and quantum gradient finding.

UCC ansatz and the hydrogen molecule

The hydrogen-molecule simulation uses a Jordan–Wigner-encoded four-qubit Hamiltonian and a singlet UCCSD ansatz built from single and double excitations. A parity-check circuit is implemented under linear nearest-neighbour connectivity.

  • UCC ansatz and the hydrogen molecule: The H2 calculation uses four molecular orbitals and maps the molecular state to qubits with the Jordan–Wigner encoding.The encoded basis distinguishes occupied and initially unoccupied spin orbitals through occupation values f_i.
  • UCC ansatz and the hydrogen molecule: The four-qubit H2 Hamiltonian contains identity, Z, ZZ, and four-qubit Pauli-string terms with coefficients h_0 through h_14.The numerical values of the Hamiltonian coefficients were obtained using OpenFermion.
  • UCC ansatz and the hydrogen molecule: The trial state is constructed with a singlet unitary coupled-cluster singles-and-doubles ansatz using excitations above the Hartree–Fock state.The UCCSD operator is Trotterized with a single Trotter step before Jordan–Wigner circuit construction.
  • UCC ansatz and the hydrogen molecule: The UCCSD operator is implemented using circuits generated from established circuit constructions, while the parity-check circuit assumes linear nearest-neighbour connectivity.The parity information is passed along the register to an ancilla and then uncomputed; this has depth O(M).

Number of measurements and error analysis

The simulations estimate energy-measurement uncertainty from repeated Hamiltonian-term measurements and allocate samples according to term strength. Extrapolation changes the standard error, while the reported combined method has comparable uncertainties to extrapolation alone.

  • Measurement allocation: Each Hamiltonian-term expectation value is estimated by repeatedly executing and measuring the circuit, with measurements allocated proportionally to term strength.The desired precision determines the total measurement count, and the standard error depends on the number of measurements and outcome variance.
  • Measurement allocation: The Pauli-string variance obeys 1 − ⟨ψ|h_j|ψ⟩^2 ≤ 1, providing an upper bound used to estimate energy-measurement error.The energy uncertainty combines the uncertainties of individual Hamiltonian terms weighted by their coefficients.
  • Extrapolation analysis: Extrapolation increases the standard error by a stretch-factor-dependent amount, with equal sample division between two points for linear extrapolation.The stretch factor used in the simulations depends on the error rate for Fig. 4 and follows a separate specification for Fig. 5.
  • Extrapolation analysis: Detection plus extrapolation produced standard errors of [0.05, 0.07, 0.04, 0.04, 0.04] mHartree, versus [0.05, 0.07, 0.04, 0.05, 0.04] mHartree for extrapolation alone.These values use the expectation-value-based standard error rather than the loose upper bound.
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