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Quantum Error Mitigation by Pauli Check Sandwiching

Alvin Gonzales, Ruslan Shaydulin, Zain Saleem, Martin Suchara

TL;DR

The paper addresses the challenge of mitigating hardware noise in near-term quantum devices without heavy quantum overhead. It introduces the Pauli Check Sandwiching (PCS) protocol, which pairs parity checks around a target circuit and uses ancilla postselection to transform the error map by eliminating anticommuting Pauli components, with both single-layer and multilayer variants. Theoretical results establish conditions for unit fidelity under restricted noise (Theorem MultilayerUnitFidelity) and practical constructions for weight-one errors, along with an efficient protocol to find suitable checks; extensive simulations on 1,850 random circuits show an average fidelity gain of 34 percentage points with six layers. The approach is flexible, composable with other methods, and applicable to subcircuits and a broad class of input states, offering a scalable path to improve fidelity in NISQ-era quantum computations with tunable overhead and without full error correction.

Abstract

We describe and analyze an error mitigation technique that uses multiple pairs of parity checks to detect the presence of errors. Each pair of checks uses one ancilla qubit to detect a component of the error operator and represents one layer of the technique. We build on the results on extended flag gadgets and put it on a firm theoretical foundation. We prove that this technique can recover the noiseless state under the assumption of noise not affecting the checks. The method does not incur any encoding overhead and instead chooses the checks based on the input circuit. We provide an algorithm for obtaining such checks for an arbitrary target circuit. Since the method applies to any circuit and input state, it can be easily combined with other error mitigation techniques. We evaluate the performance of the proposed methods using extensive numerical simulations on 1,850 random input circuits composed of Clifford gates and non-Clifford single-qubit rotations, a class of circuits encompassing most commonly considered variational algorithm circuits. We observe average improvements in fidelity of 34 percentage points with six layers of checks.

Quantum Error Mitigation by Pauli Check Sandwiching

TL;DR

The paper addresses the challenge of mitigating hardware noise in near-term quantum devices without heavy quantum overhead. It introduces the Pauli Check Sandwiching (PCS) protocol, which pairs parity checks around a target circuit and uses ancilla postselection to transform the error map by eliminating anticommuting Pauli components, with both single-layer and multilayer variants. Theoretical results establish conditions for unit fidelity under restricted noise (Theorem MultilayerUnitFidelity) and practical constructions for weight-one errors, along with an efficient protocol to find suitable checks; extensive simulations on 1,850 random circuits show an average fidelity gain of 34 percentage points with six layers. The approach is flexible, composable with other methods, and applicable to subcircuits and a broad class of input states, offering a scalable path to improve fidelity in NISQ-era quantum computations with tunable overhead and without full error correction.

Abstract

We describe and analyze an error mitigation technique that uses multiple pairs of parity checks to detect the presence of errors. Each pair of checks uses one ancilla qubit to detect a component of the error operator and represents one layer of the technique. We build on the results on extended flag gadgets and put it on a firm theoretical foundation. We prove that this technique can recover the noiseless state under the assumption of noise not affecting the checks. The method does not incur any encoding overhead and instead chooses the checks based on the input circuit. We provide an algorithm for obtaining such checks for an arbitrary target circuit. Since the method applies to any circuit and input state, it can be easily combined with other error mitigation techniques. We evaluate the performance of the proposed methods using extensive numerical simulations on 1,850 random input circuits composed of Clifford gates and non-Clifford single-qubit rotations, a class of circuits encompassing most commonly considered variational algorithm circuits. We observe average improvements in fidelity of 34 percentage points with six layers of checks.
Paper Structure (13 sections, 3 theorems, 26 equations, 17 figures, 3 algorithms)

This paper contains 13 sections, 3 theorems, 26 equations, 17 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Methods
  4. Pauli Sandwich Error Mitigation Protocol: Single Layer
  5. Errors Detected by the Pauli Sandwich
  6. Pauli Sandwich Error Mitigation Protocol: Multiple Layers
  7. Upper Bounds on Fidelity and Required Number of Checks
  8. General Errors and Hardware Considerations
  9. Protocol for Finding Checks Quickly
  10. Numerical Results
  11. Conclusions
  12. Data Availability
  13. Code Availability

Key Result

Theorem 1

If errors are restricted to act only on the compute qubits, for any noisy unitary quantum circuit $U$ acting on $n$ compute qubits, there exist checks (see Fig. fig:multilayerNoisy) such that the fidelity between the post selected state and a noiseless run (noiseless execution of U only) reaches one

Figures (17)

  • Figure 1: Overview of the one-layer version of the PCS scheme. $U$ represents the gates of the computation and acts across $n$ compute qubits. $U$ is sandwiched between two controlled unitaries comprising $\tilde{C}_1$ and $\tilde{C}_2$ that satisfy Eq. \ref{['eq:conditionOnChecks1']}. The ancilla is the bottom qubit. The measurement is performed in the $\{\ket{0}, \ket{1}\}$ basis. The measurement outcome one is discarded, and zero is kept.
  • Figure 2: Noisy single-layer scheme
  • Figure 3: Multilayer scheme. There are $n$ compute qubits, $m$ layers, and $m$ ancillas. The second index in the controlled unitaries represents the layer. Each layer uses one ancilla and two checks. The checks sandwich the input circuit.
  • Figure 4: Noisy multilayer scheme
  • Figure 5: Example of checks that detect all errors. The upper bound on fidelity is saturated at four layers for this randomly generated Clifford input circuit consisting of two qubits and 30 cnot gates. We use depolarizing noise for the given noise model in Fig. \ref{['fig:multilayerNoisy']}. The two-qubit error rate is ten times the single-qubit error rate. The single qubit-error rate ranges from $10^{-5}$ to $10^{-1}$. At $10^{-1}$, each cnot gate (acting on the compute qubits only) is followed by a two-qubit maximal depolarizing channel. Regardless, the postselected state is noiseless, as predicted by Theorem \ref{['thm:MultilayerUnitFidelity']}.
  • ...and 12 more figures

Theorems & Definitions (7)

  • Theorem 1: Unit Fidelity
  • proof
  • Proposition 1: Weight-One Kraus Operators: Two layers of max weight checks are sufficient
  • proof
  • Remark
  • Proposition 2: Any Error: $2n$ number of weight-one checks are sufficient
  • proof