Exponential Error Suppression for Near-Term Quantum Devices

Bálint Koczor

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Exponential Error Suppression for Near-Term Quantum Devices

Bálint Koczor

arXiv:2011.05942v4Nov 11, 2020quant-ph

TL;DR

Current NISQ devices make quantum error correction impractical, while existing mitigation methods generally lack exponential suppression. The paper introduces Error Suppression by Derangement, showing that independently prepared copies bridged by a shallow derangement circuit can exponentially suppress expectation-value errors.

Problem
NISQ devices make quantum error-correcting codes impractical, while existing mitigation approaches do not generally provide exponential error suppression for expectation-value estimation.
Method
Error Suppression by Derangement prepares independent copies and applies a permutation derangement that allows only permutation-symmetric states to contribute to expectation-value measurements.
Results
The approach suppresses expectation-value errors exponentially with the number of copies and remains compatible with independently prepared states bridged by a shallow circuit.
Takeaways & Limitations
ESD offers a NISQ-compatible route to exponential expectation-value error suppression without integrating error correction into the main computation.
Takeaways & Limitations
Coherent errors limit ESD precision, although the authors report that their impact decreases as computational scale increases.

Abstract

from arXiv · show

As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level $E$ at a cost in qubit-count $n$ that is merely poly-logarithmic in $1/E$. However in the NISQ era, the complexity and scale required to adopt even the smallest QEC is prohibitive. Instead, error mitigation techniques have been employed; typically these do not require an increase in qubit-count but cannot provide exponential error suppression. Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression. We introduce the Error Suppression by Derangement (ESD) approach: by increasing the qubit count by a factor of $n\geq 2$, the error is suppressed exponentially as $Q^n$ where $Q<1$ is a suppression factor that depends on the entropy of the errors. The ESD approach takes $n$ independently-prepared circuit outputs and applies a controlled derangement operator to create a state whose symmetries prevent erroneous states from contributing to expected values. The approach is therefore `NISQ-friendly' as it is modular in the main computation and requires only a shallow circuit that bridges the $n$ copies immediately prior to measurement. Imperfections in our derangement circuit do degrade performance and therefore we propose an approach to mitigate this effect to arbitrary precision due to the remarkable properties of derangements. a) they decompose into a linear number of elementary gates -- limiting the impact of noise b) they are highly resilient to noise and the effect of imperfections on them is (almost) trivial. In numerical simulations validating our approach we confirm error suppression below $10^{-6}$ for circuits consisting of several hundred noisy gates (two-qubit gate error $0.5\%$) using no more than $n=4$ circuit copies.

I. INTRODUCTION … A. Noisy Quantum States and Entropies

The paper introduces Error Suppression by Derangement (ESD), which targets expectation-value estimation by using multiple state copies and permutation symmetry to achieve exponential error suppression without full QEC. Its NISQ-oriented design trades increased qubit count for modular preparation and a shallow bridging circuit, while efficacy depends on the noisy state's error distribution and entropies.

I. INTRODUCTION: At least 2 copies are required, making ESD more hardware-expensive than many NISQ mitigation schemes but enabling exponential suppression unavailable to other NISQ solutions.The approach therefore occupies an intermediate position between NISQ mitigation and full QEC.
A. Estimating Expectation Values: Without comprehensive error correction, state-preparation errors bias observable estimates, while existing mitigation methods require more measurements and circuit variants.These methods can reduce errors without increasing qubit count but address measurement errors rather than providing QEC-like protection.
A. Estimating Expectation Values: ESD achieves exponential error suppression for expectation-value estimation by preparing multiple copies and protecting collective permutation symmetry with derangement operators.Most preparation errors break permutation symmetry and are effectively filtered out.
A. Estimating Expectation Values: Expectation-value estimation is central to near-term applications, including variational quantum eigensolvers for quantum chemistry and materials science.Hamiltonian expectation values are decomposed into sums of Pauli-operator expectation values.
A. Estimating Expectation Values: ESD remains NISQ-friendly because copies are prepared independently, while the derangement circuit bridges them immediately before measurement and uses a linear number of primitive gates.The shallow bridge is intended to accumulate less noise than state preparation and can be combined with other mitigation techniques.
II. PRELIMINARIES: Noisy devices prepare mixed states whose dominant eigenvector need not equal the ideal computed state, even under purely incoherent error models.The coherent mismatch can nevertheless be exponentially smaller than accumulated erroneous contributions according to cited theoretical guarantees.
A. Noisy Quantum States and Entropies: λ = 10^-6 is allowed provided it remains the dominant component, although sampling costs may become prohibitive at extremely low λ.The dominance condition is λ > (1−λ)p_k for every erroneous eigenvalue contribution.
A. Noisy Quantum States and Entropies: Rényi entropies of the erroneous-state probability distribution have a crucial effect on ESD efficacy, and typical experimental systems may have large H_n(p).The discussion does not restrict the error probability distribution p.

B. Main Idea · III. RESULTS

The method suppresses bias from erroneous eigenvectors when estimating expectation values by measuring observables combined with register permutations. Derangements exploit eigenvector orthogonality so that only permutation-symmetric states contribute.

B. Main Idea: Expectation-value estimates are biased when erroneous eigenvectors contribute terms such as ⟨ψk|σ|ψk⟩.The target is the ideal-state expectation value ⟨ψid|σ|ψid⟩, approximated in practice using the dominant eigenvector.
B. Main Idea: Measuring σ SWAP1n changes the register ordering when one register contains an orthogonal erroneous eigenvector.The swap exchanges registers 1 and n before measuring the expectation value.
B. Main Idea: 0 results for the swapped erroneous component because ⟨ψk|ψ⟩ = 0 by eigenvector orthogonality.The matrix element factors as ⟨ψk|σ|ψ⟩⟨ψk|ψ⟩.
B. Main Idea: Derangements generalize the swap by permuting all registers, allowing only permutation-symmetric states to contribute to expectation values.The operation is defined as a permutation of the register ordering and is presented as completely general.
B. Main Idea: The register permutation changes |ψk, ψ, . . . ψ⟩ to |ψ, ψ, . . . ψk⟩ before the orthogonality-based cancellation.This reordered state is the mechanism producing the zero contribution.
B. Main Idea: Fig. 1 presents the derangement construction while leaving room for different physical implementations.Alternative implementations are discussed later in the paper.

A. Exponential Error Suppression

The derangement measurement exponentially suppresses erroneous contributions when multiple identical circuit outputs are combined. Its suppression factor depends on the error distribution’s Rényi entropy, while Methods A and B estimate the target expectation value with exponentially decaying errors.

Derangement measurement: Only permutation-symmetric combinations pass the derangement measurement, suppressing erroneous contributions exponentially as the number n of copies increases.The circuit measures the expectation value of σD_n, where D_n permutes the n input registers.
Approximation methods: Both Methods A and B estimate ⟨ψ|σ|ψ⟩ through ancilla probability prob0, and their approximation errors E_A and E_B generally decay exponentially.Method B assumes knowledge of the dominant eigenvalue λ, whereas Method A additionally estimates prob′0 without the controlled-σ gate.
Exponential suppression: Q < 1 bounds the error sequence Q^n, establishing exponential suppression with the number of copies.The bound has the form Q^n ≤ const × Q^n, with the suppression factor Q < 1.
Error-distribution dependence: Q^n = (λ^-1−1)^n exp[(1−n)H_n(p)] links suppression directly to the Rényi entropy H_n(p) of the error distribution.The same bounds extend to unit-norm observables that are linear combinations of Pauli strings.

B. Numerical Simulations · C. Effect of Non-Identical States

Numerical simulations test the suppression bounds on a noisy 12-qubit variational circuit, while non-identical copies retain exponentially decreasing approximation errors when sharing a dominant eigenvector with λmin > 1/2. Commuting, perturbed copies show approximation errors similar to the identical-copy case and remain approximately bounded by Result 1.

B. Numerical Simulations: A 12-qubit variational circuit with 10 alternating layers and 372 quantum gates provides the simulation setting.The circuit models a noisy computation typically used in variational quantum algorithms.
B. Numerical Simulations: The noise model applies 0.5% depolarising noise to each two-qubit gate and 0.05% to each single-qubit gate.The resulting state has dominant eigenvalue λ ≈0.51 and Renyi entropies H2(p) = 4.69, H3(p) = 4.38, H4(p) = 4.23, and H∞(p) = 3.63.
B. Numerical Simulations: Despite incoherent noise, the dominant eigenvector |ψ⟩ differs slightly from the error-free computation, so Fig. 2 computes errors using the noisy state’s dominant eigenvector.This defines the eigenvector used for the numerical error evaluation.
B. Numerical Simulations: The simulated error bounds use 500 Pauli-string observables and compare entropy-based bounds with Q^n ≤ (λ^-1−1)^n(pmax)^(n−1).Here pmax = 0.026 and Q = 0.026, with Method A and Method B shown separately.
B. Numerical Simulations: Method B slightly outperforms Method A but requires exact or very precise knowledge of the dominant eigenvalue λ.Existing approaches may determine λ precisely in special cases.
C. Effect of Non-Identical States: For arbitrarily different copies sharing dominant eigenvector |ψ⟩, Result 2 gives exponentially decreasing approximation errors when the worst copy has λmin > 1/2.The copies satisfy ρ1 ≠ ρ2 ≠ . . . ρn while retaining the common dominant eigenvector.
C. Effect of Non-Identical States: When copies commute, perturbed states with ∥ρk − ρl∥≈10^-2 produce approximation errors similar to the identical-copy simulation and approximately bounded by Result 1.The commuting case has common eigenvectors but different eigenvalues and follows an effective sequence Qeff.

D. Complexity Analysis · E. Derangements of Quantum Registers

The ESD resource cost is logarithmic in the inverse target precision for state copies, while measurement sampling can incur a higher polynomial cost governed by λ and Q. Its derangement circuits require linearly many controlled-SWAP gates and can be selected to accommodate hardware constraints.

D. Complexity Analysis: n = O(ln E−1/ ln Q−1) copies suffice to reach precision E in estimating ⟨ψ|σ|ψ⟩, with Q < 1 determined by Rényi entropies.The total qubit requirement is nN + 1, where N is the number of qubits in the computational state ρ.
D. Complexity Analysis: f = ln[λ−1/ ln Q−1] increases the polynomial order beyond the standard shot-noise limit O(E−2).A general upper bound on f is derived in Lemma 2.
D. Complexity Analysis: Ns grows polynomially with the inverse target precision E−1, while the required system size through n grows logarithmically with E−1.The complexity depends on the largest eigenvalue λ and suppression factor Q.
E. Derangements of Quantum Registers: Derangements permute n ordered quantum registers so that no element remains in its original position, generalizing SWAP operations between register subspaces.For n = 3, the constructions include cyclic register mappings such as |ψ1, ψ2, ψ3⟩ → |ψ3, ψ1, ψ2⟩ and |ψ2, ψ3, ψ1⟩.
E. Derangements of Quantum Registers: (n−1)! distinct derangement operators exist in general, and any one construction suffices for the ESD scheme.For n = 4 there are 6 possibilities; cyclic shifts are straightforward, while alternatives can account for hardware connectivity.
E. Derangements of Quantum Registers: N(n −1) elementary controlled two-qubit SWAP gates implement derangement operators efficiently for N-qubit registers and n copies.These minimal circuits optimally implement derangement operators and can be constructed by mapping permutations to graph trees.
E. Derangements of Quantum Registers: O(N) derangement-circuit gates have diminishing relative significance when preparing |ψ⟩ requires O[a(N)N] gates and a(N) generally grows with circuit depth.Practical problems are generally expected to require more than constant-depth circuits, so the main computation can grow faster than O(N).

IV. NOISE ROBUSTNESS AND LIMITATIONS · A. Mitigating Experimental Imperfections

Derangement measurements remain highly resilient to experimental imperfections, with errors often attenuating the output probability nearly linearly. Extrapolation can substantially mitigate this noise, although controlled-SWAP noise accumulates as qubit count increases.

A. Mitigating Experimental Imperfections: Derangement operators protect permutation symmetry even under experimental noise and are highly resilient to experimental imperfections.This resilience is established generally in Example 3.
A. Mitigating Experimental Imperfections: 10^-2: unmitigated errors in determining prob0 remain below this value for all 50 randomly selected states.The simulation uses 13 qubits, comprising 3 copies of a 4-qubit state.
A. Mitigating Experimental Imperfections: 3 × 10^-3: elementary controlled-SWAP gates undergo 3-qubit depolarisations with this probability in the simulated circuit.The circuit contains 13 qubits and three copies of a 4-qubit state.
A. Mitigating Experimental Imperfections: Most derangement-measurement errors nearly linearly attenuate prob0 and can in principle be corrected by extrapolating prob0(ϵ) to zero noise.The approach estimates prob0(ϵ) at different ϵ values and extrapolates to ϵ = 0.
A. Mitigating Experimental Imperfections: ν+1: determining prob0(ϵ) at this many different ϵ values permits fitting a degree-ν polynomial and recovering the ideal probability in principle.Theorem 3 describes expectation values through prob0(ϵ) = Pν.
A. Mitigating Experimental Imperfections: Extrapolation error decreases exponentially with the degree of the fitted polynomial in the simulated mitigation results.This behavior is demonstrated using blue-circle results in Fig. 3.
A. Mitigating Experimental Imperfections: Controlled-SWAP noise accumulates as the number of qubits increases, limiting the mitigation approach despite extrapolation substantially reducing derangement-circuit noise.The paper identifies increasing controlled-SWAP noise as a limitation in realistic scenarios.

B. Limitations of the Technique · V. PRACTICAL APPLICATIONS

The approach is limited mainly by coherent and measurement errors, although established mitigation methods and favorable behavior in variational algorithms can reduce these effects. A practical spin-ring demonstration shows useful error suppression with shallow variational circuits, while measurement overhead and coherent mismatch constrain performance.

B. Limitations of the Technique: Coherent errors limit ESD precision because the approach is oblivious to them, although Pauli twirling can convert them into incoherent errors.The impact of coherent mismatch decreases with computation scale, with theoretical guarantees and an Appendix mitigation procedure provided.
B. Limitations of the Technique: Variational quantum algorithms are especially suitable because coherent mismatch is quadratically smaller for eigenstate preparation and variational optimization implicitly reduces its impact.Variational parameters can also be slightly re-adjusted to address this error.
B. Limitations of the Technique: Measurement errors are neglected in the analysis, but established techniques exist to mitigate bias in the probability of collapsing into state 0.
V. PRACTICAL APPLICATIONS: Shallow circuits can still approximate ground-state energies of Hamiltonians unavailable to classical methods, motivating practical NISQ applications.The spin-ring Hamiltonian is relevant to many-body localization, has a simple structure with linearly scaling Pauli observables, and relates to QAOA and spin systems.
V. PRACTICAL APPLICATIONS: 94% of the noise was assumed amplifiable, while approximately 6% depolarising noise limited extrapolation but ESD remained effective under arbitrary noise models.The comparison models dephasing and damping as perfectly amplifiable and depolarising noise as unamplifiable.
V. PRACTICAL APPLICATIONS: 20 ansatz layers and N = 6 qubits with n = 2 noisy copies yielded a noise-free approximation error of ΔE ≈10^-4 in the simulated application.The circuit uses a variational Hamiltonian ansatz, with hardware-native single-qubit Ry and Rz rotations and XX entangling gates considered.
V. PRACTICAL APPLICATIONS: ξ ⪅1 is the practically important region where errors can fall below shot noise and the ansatz approximation error, whereas ξ ⪆2 is inaccessible because measurement overhead rises rapidly.As n →∞, a constant error remains because of coherent mismatch; the overhead scales as f = O(ξ) when λ = O(e^-ξ).
V. PRACTICAL APPLICATIONS: N > 20 qubits and deeper ansätze are expected to improve practical value by increasing the main-computation-to-derangement gate ratio, while larger computations reduce coherent mismatch.The demonstration is characterized as a worst-case scenario, and more complex Hamiltonians may require faster-growing gate counts.

VI. DISCUSSION AND CONCLUSION

The discussion presents ESD as a symmetry-based method that exponentially suppresses expectation-value errors by bridging independently prepared copies with a shallow derangement circuit. It also identifies limitations, distinguishes ESD from related SWAP-based work, and notes broad implementation possibilities.

Contributions: ESD suppresses expectation-value errors exponentially in the number n of identical copies while requiring only a shallow circuit immediately before measurement.The copies can be prepared independently and are bridged only by the derangement circuit.
Limitations: ESD cannot address coherent noise or a coherent mismatch in the dominant eigenvector.These errors can nevertheless be exponentially smaller than incoherent fidelity decay and decrease as computation scale increases.
Limitations: With a perfect derangement circuit, ESD has sample complexity polynomial in E^-1 and comparable to the standard shot-noise limit.This characterization applies under the stated perfect-circuit assumption.
Relation to prior work: For n = 2 copies, ESD is comparable to a modification of the usual SWAP-test circuit, but it targets the dominant computational eigenvector rather than the input mixed state.Prior copy-based approaches applied permutation symmetries to tasks including spectral reconstruction, entanglement probing, quantum software, and state discrimination.
Outlook: The analyzed circuit is only one realization of a general ESD principle, leaving substantial room for alternative physical implementations and circuit improvements.The example circuit also possesses many invariants that may support further optimization.
Resources: Simulation and demonstration materials are available in the online repository.The repository is cited as references.

Appendix A: Derangement measurements and suppressing errors

The appendix derives how derangement measurements estimate expectation values and shows that ESD suppresses errors exponentially in the error distribution’s Rényi entropy. High-entropy errors can yield dramatic suppression, whereas zero-entropy errors can make the approach fail or substantially weaken its performance.

Exponential suppression: Error probabilities are suppressed exponentially as p_k^n, while the dominant contribution is attenuated by λ^n.The appendix’s expectation-value construction produces exponential suppression of erroneous eigenstate contributions, with a corresponding attenuation of the dominant eigenvalue term.
High-entropy errors: 640000-times smaller error is achieved for n = 3 with λ = 0.8 and a uniform high-entropy error over 100 eigenvectors.The bound gives |E| ≤ 8×10^-7 for Tr[ρ^3σ] = 0.512⟨ψ|σ|ψ⟩ + E, while the error distribution has high Rényi entropy.
High-entropy errors: 640000-times smaller error yields the estimate 0.512001 for n = 3, closely approximating the ideal value 0.512.Method A estimates the dominant eigenvalue from measurements using the identity observable, while Method B assumes it is known precisely.
Zero-entropy errors: 0-entropy error distributions cause complete breakdown when λ ≤ 1/2.For ρ = λ|ψ⟩⟨ψ| + (1 −λ)|ψerr⟩⟨ψerr|, the error distribution has p2 = 1 and pk = 0 for k > 2.
Zero-entropy errors: 64 suppression is obtained for n = 3 in the zero-entropy case, versus 640000 suppression for the high-entropy example.The resulting expression is Tr[ρ^3σ] = 0.512⟨ψ|σ|ψ⟩ + 0.008⟨ψerr|σ|ψerr⟩, demonstrating the dependence on Rényi entropy.

Appendix B: Exponentially decreasing upper bounds on approximation errors

Appendix B proves that ESD approximation errors decrease exponentially with the number of copies, governed by the suppression factor Q<1 and the error distribution’s Rényi entropy. Achieving precision E therefore requires only logarithmically many copies, while measurement costs remain polynomial in E^-1.

Theorem 2: Theorem 2 bounds the approximation errors as |E_A| ≤ 2Q^n/(1+Q^n) and |E_B| ≤ Q^n.Method A estimates both Tr[ρ^nσ] and Tr[ρ^n], whereas Method B assumes the largest eigenvalue λ is known.
Lemma 1: Q_n=(λ^-1−1)^n exp[−(n−1)H_n(p)] decreases exponentially with Rényi entropy H_n(p), and Q_n≤(p_max)^−1Q^n for Q:=(λ^-1−1)p_max<1.Here p_max is the largest error probability.
Measurement cost: The overall Method A measurement cost is N_s=O[E^-2(1+2f)], while Method B requires polynomially many samples in E^-1.The polynomial order is determined by f:=ln(λ^-1)/ln(Q^-1).
Measurement cost: f≤1.26 when λ=0.6, while f≤0.16 for λ≥0.9, showing that good-quality states incur only small polynomial overhead.The bound can be smaller for higher-entropy probability distributions.

Appendix C: Effect of violating assumptions

The ESD error suppression remains effective when copies are non-identical, provided they share a dominant eigenvector. For arbitrary such copies, the error scales as O([λ_min−1−1]^n) when λ_min > 1/2, while commuting copies admit an effective suppression sequence Q_eff.

Non-identical copies: Non-identical states with an identical dominant eigenvector still yield exponentially decreasing error bounds.This extends the main result beyond perfectly identical copies.
Commuting copies: For commuting copies, the error bounds approximately retain Theorem 2’s form through an effective sequence Q_eff determined by the smallest dominant eigenvalue λ_min.Method B assumes the dominant eigenvalues are known, and the error depends on λ_min.
Commuting copies: Method A satisfies |E_A| ≤ 2Q_eff^n/(1 + Q_eff^n) for commuting non-identical copies.The bound is obtained by upper-bounding the relevant product using λ_min and the effective sequence Q_eff.
Commuting copies: Q_eff is not guaranteed to be below 1 in general, but high-entropy probability distributions and n > 1 are expected to preserve similar exponential error decay.This is a limitation of the generalized commuting-copy analysis.

1. Coherent mismatch in incoherent error channels

Incoherent error channels can slightly mismatch the dominant eigenvector, but the mismatch remains small in relevant regimes. As circuit depth increases, incoherent decay becomes more damaging than the linearly increasing coherent mismatch.

Limitation: ESD cannot address coherent errors arising when the density matrix’s dominant eigenvector is a superposition of ideal and erroneous states.Such errors are expected from systematic effects such as miscalibrated rotation angles.
Incoherent-channel mismatch: For incoherent noise, eigenvectors can mismatch unless the ideal and error states commute; single-qubit depolarisation is a commuting exception, unlike general multi-qubit cases.The relevant condition is [ρ, ρerr] = 0.
Magnitude of mismatch: 10^-4: numerical simulations found the dominant eigenvector’s infidelity below 10^-4 relative to the noise-free state.High-entropy error probabilities ensure the commutator norm is much smaller than 1, keeping the coherent mismatch small.
Scaling: O(ϵ): perturbation theory bounds the eigenvector correction’s scaling by O(ϵ), while the coherent mismatch in the investigated simulations scales as O(ϵ2).The correction is bounded through the norm of a column of ρerr.
Circuit-depth dependence: Linearly: coherent mismatch grows with gate count, whereas the dominant eigenvalue decays exponentially as (1−ϵ)ν, making incoherent decay more damaging.The ratio η2/η1 also appears to decrease exponentially with the number of gates in the investigated region.

a. Mitigating the coherent mismatch · Appendix D: Noise resilience of derangements and error extrapolation

The paper mitigates coherent mismatch through low-order polynomial extrapolation, while Appendix D shows that derangement measurements preserve permutation-symmetry protection under many derangement-circuit errors. Remaining errors affecting the measured register cause nontrivial degradation, but linear or polynomial extrapolation can recover ideal expectation values under stated conditions.

a. Mitigating the coherent mismatch: A 12-qubit simulation uses n = 3 copies to evaluate the ideal-state contribution Tr[ρn] from Method A of Theorem 2.The ideal state |ψid⟩ is the result of a perfect, noise-free circuit evaluation.
a. Mitigating the coherent mismatch: Low-order polynomial fitting suppresses coherent mismatch below the Result 1 error bound, with increasing degree eventually reaching the bound’s saturation level.The extrapolation varies state-preparation gate error rather than derangement-process error.
a. Mitigating the coherent mismatch: η2/η1 decays exponentially with gate count at fixed two-qubit gate error ϵ = 10−2, making coherent mismatch negligible for large systems.The figure describes η1 as fidelity loss from coherent mismatch and η2 as fidelity decay from incoherent noise.
Appendix D: Noise resilience of derangements and error extrapolation: Derangement errors preserve the orthogonality relations that exclude non-symmetric input combinations from contributing to the output.This establishes resilience even when random errors affect the derangement procedure.
Appendix D: Noise resilience of derangements and error extrapolation: Derangement-process errors reduce error-free output probabilities only linearly through the 1 −ϵ factor, and linear extrapolation can completely correct this attenuation.For symmetric inputs, errors on registers other than the observable’s register cancel while preserving the correct contribution.
Appendix D: Noise resilience of derangements and error extrapolation: Errors affecting the observable’s register degrade the result nontrivially through ⟨U1ψ|σ|U1ψ⟩, unlike errors on other registers.The derangement measurement otherwise protects permutation symmetry despite experimental noise.
Appendix D: Noise resilience of derangements and error extrapolation: ν noisy gates make any expectation value a degree ν polynomial in error probability, enabling exact recovery of E(0) from ν+1 error-rate points.Lagrange or Newton interpolation provides explicit reconstruction formulas; Padé fitting offers an approximation requiring experimentally fitted coefficients.
Appendix D: Noise resilience of derangements and error extrapolation: Padé-based approximation relies on ηk ≈ 0, an assumption motivated by most derangement errors not contributing directly to the output.When this condition is not exact, the coefficients can instead be fitted to experimental data.

Appendix E: Hardware-native implementation of derangement circuits · 1. Recompiling controlled-SWAP gates

The appendix develops hardware-native recompilations of controlled-SWAP gates for derangement circuits, exploiting full, local SU(4), and observable-aware equivalences. Depending on the native gateset and recompilation type, implementations use six or five two-qubit entangling gates, or substantially fewer native three-qubit gates.

1. Recompiling controlled-SWAP gates: Controlled-SWAP gates are decomposed into elementary two-qubit SWAP operations and recompiled for hardware-native gatesets.The recompilation targets the elementary controlled-SWAP gates appearing in register permutations.
1. Recompiling controlled-SWAP gates: Local SU(4)-equivalent recompilation is valid when subsequent operations permit local transformations without changing the ancilla measurement outcome.Fully equivalent recompilation remains necessary when later controlled-SWAP operations act on the same registers, while local equivalence can be used for terminal operations.
1. Recompiling controlled-SWAP gates: The final Rz rotation on the control qubit can be removed because it commutes with controlled-SWAP and can merge with the ancilla’s pre-measurement basis transformation.This optimization applies to the circuits shown for hardware-native recompilation.
1. Recompiling controlled-SWAP gates: 6 two-qubit operations generally implement an elementary controlled-SWAP gate in the fully equivalent case.The appendix also distinguishes observable-aware recompilation, where the controlled-SWAP and controlled observable can be recompiled together.
1. Recompiling controlled-SWAP gates: Native three-qubit gates yield compact controlled-SWAP circuits, including a decomposition using a single CC[P] gate plus 2(1) controlled-Z rotation gates.The count 2(1) corresponds to the fully (locally) equivalent recompilation cases.
1. Recompiling controlled-SWAP gates: 6 (5) XX-rotation applications implement controlled-SWAP under fully (locally) equivalent recompilation.The assumed hardware natively provides single-qubit Y and Z rotations and a two-qubit XX gate.
1. Recompiling controlled-SWAP gates: 6 (5) parametrised-SWAP entangling-gate applications are still required for fully (locally) equivalent recompilation.Although parametrised SWAP is more general and can generate SWAP at special angles, it does not significantly improve the controlled-SWAP recompilation.
1. Recompiling controlled-SWAP gates: When XX and XXX gates are available, the controlled-SWAP can be expressed through commuting terms mapped to native multi-qubit gates and single-qubit rotations.The resulting series includes zxx, zyy, zzz, xx, yy, zz, and an ancilla z gate that can be removed before measurement.

2. Exploiting symmetries in derangement circuits … 3. Ground state simulation in Fig. 4

The paper exploits derangement-circuit symmetries through randomized Pauli twirling, constrained permutations, and related constructions, while numerical studies examine ESD and extrapolation across progressively richer noisy simulations. In the six-qubit ground-state example, ESD is evaluated against zero-noise extrapolation under a physically motivated noise model.

2. Exploiting symmetries in derangement circuits: Derangement circuits offer many invariants and permutation patterns that can be exploited through connectivity-aware, randomized, or twirling-inspired constructions.Nearest-neighbour swaps support constrained derangements, while arbitrary connectivity permits selecting or randomizing among derangement patterns.
2. Exploiting symmetries in derangement circuits: Randomly applying Pauli strings before and after controlled derangements preserves ideal expectation values while homogenising derangement-circuit errors.The post-derangement Pauli operation relabels register indices according to the implemented permutation, analogously to twirling.
2. Exploiting symmetries in derangement circuits: 30-50% of errors are reduced by generalized twirling in simulations of 50 randomly selected observables.The reported reduction applies despite additional noise from the controlled Pauli strings.
1. Simulations in Fig. 2: 372 noisy gates are used in the 12-qubit, 10-layer alternating-layer simulation, with 0.5% two-qubit and 0.05% single-qubit depolarising-error probabilities.Approximation errors are measured relative to the dominant eigenvector expectation value; coherent mismatch is below 10^-4.
2. Simulations in Fig. 3: n = 3 copies of a noisy 4-qubit state are simulated with controlled-SWAP depolarising noise probability 10^-3 across 50 randomly parameterised instances.Extrapolation techniques are applied separately to each randomly generated parameter set.
3. Ground state simulation in Fig. 4: The ground-state study fixes N = 6 qubits and targets a precision of ∆E = 10^-4 using an alternating variational Hamiltonian ansatz.Parameters are optimized through 1000 natural-gradient iterations over 5 independent optimisations.
3. Ground state simulation in Fig. 4: The ground-state noise model combines amplifiable dephasing, damping at 0.1ϵ, and non-amplifiable depolarising noise at 0.07ϵ, with two-qubit probabilities scaled as ϵ →5ϵ.The model is intended to capture finite-T2, T1 relaxation, and imperfect-control effects, while theoretical bounds depend only on the density-matrix eigenvalue distribution.
3. Ground state simulation in Fig. 4: Close to 85% of errors are reduced by polynomial zero-noise extrapolation when the circuit error rate satisfies ξ < 1.Polynomial fitting slightly outperformed the tested exponential and linear fits; the comparison also includes the noisy derangement circuit.