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Error mitigation for partially error-corrected quantum computers

Ben DalFavero, Ryan LaRose

TL;DR

It is shown how logical ancilla qubits can arbitrarily reduce the sampling complexity of error cancellation in a continuous space-time tradeoff, and conjecture that any error mitigation protocol with (sub-)polynomial sample complexity requires exponential time and/or space, even when logical qubits are utilized as a resource.

Abstract

We present a method for quantum error mitigation on partially error-corrected quantum computers - i.e., computers with some logical qubits and some noisy qubits. Our method is inspired by the error cancellation method and is implemented via a circuit for convex combinations of channels which we introduce in this work. We show how logical ancilla qubits can arbitrarily reduce the sampling complexity of error cancellation in a continuous space-time tradeoff, in the limiting case achieving $O(1)$ sample complexity which circumvents lower bounds for sample complexity with all known error mitigation techniques. This comes at the cost of exponential circuit depth, however, and leads us to conjecture that any error mitigation protocol with (sub-)polynomial sample complexity requires exponential time and/or space, even when logical qubits are utilized as a resource. We anticipate additional applications for our quantum circuits to implement convex combinations of channels, and to this end we discuss one application in simulating open quantum systems, showing an order of magnitude reduction in gate counts relative to current state-of-the-art methods for a canonical problem.

Error mitigation for partially error-corrected quantum computers

TL;DR

It is shown how logical ancilla qubits can arbitrarily reduce the sampling complexity of error cancellation in a continuous space-time tradeoff, and conjecture that any error mitigation protocol with (sub-)polynomial sample complexity requires exponential time and/or space, even when logical qubits are utilized as a resource.

Abstract

We present a method for quantum error mitigation on partially error-corrected quantum computers - i.e., computers with some logical qubits and some noisy qubits. Our method is inspired by the error cancellation method and is implemented via a circuit for convex combinations of channels which we introduce in this work. We show how logical ancilla qubits can arbitrarily reduce the sampling complexity of error cancellation in a continuous space-time tradeoff, in the limiting case achieving sample complexity which circumvents lower bounds for sample complexity with all known error mitigation techniques. This comes at the cost of exponential circuit depth, however, and leads us to conjecture that any error mitigation protocol with (sub-)polynomial sample complexity requires exponential time and/or space, even when logical qubits are utilized as a resource. We anticipate additional applications for our quantum circuits to implement convex combinations of channels, and to this end we discuss one application in simulating open quantum systems, showing an order of magnitude reduction in gate counts relative to current state-of-the-art methods for a canonical problem.

Paper Structure

This paper contains 1 section, 3 theorems, 16 equations, 2 figures, 1 table.

Table of Contents

  1. More details on gate counts for Rabi flopping

Key Result

Theorem 1

Let $\mathcal{C} := \mathcal{U}_L \ \cdots \ \mathcal{U}_1$ be an $n$-qubit, depth $L$ quantum circuit preparing the state $\rho$, and let $A$ be an observable. Provided $O(k)$ logical qubits, there exists an error cancellation protocol to estimate $\Tr [ \rho A]$ with negativity $O(\gamma^{L - k})$

Figures (2)

  • Figure 1: A quantum circuit to implement the convex combination of channels \ref{['eqn:ccc']}. The procedure is similar to the LCU technique with unitary dilations $U_{\mathcal{E}_\alpha}$ for each channel $\mathcal{E}_\alpha$, however there is no post-selection and the preparation is deterministic. The unitary $V$ prepares the probability distribution $V|0\rangle = \sum_\alpha \sqrt{p_\alpha} |\alpha\rangle$ and gates labeled $|\alpha\rangle$ on the top register represent operations controlled on the state $|\alpha\rangle$. The second register used for channel dilations requires $\log M$ qubits, where $M = \max_\alpha M_\alpha$ and $M_\alpha$ is the number of Kraus operators in channel $\mathcal{E}_\alpha$. At the end of the circuit, the bottom register has the convex combination of channels applied to the input state $\rho$, i.e. $\sum_\alpha p_\alpha \mathcal{E}_\alpha ( \rho )$, as proved in the main text.
  • Figure 2: A unitary circuit to implement the amplitude damping channel with input state $\rho$ and output state $\rho'$. The damping parameter is $\beta = \sin^2(\theta)$Nielsen_Chuang_2010.

Theorems & Definitions (5)

  • Theorem 1: Error cancellation on partially error-corrected quantum computers
  • Corollary 1: Constant-sample error cancellation on partially error-corrected quantum computers, at the cost of exponential depth
  • Theorem 2: Quantum circuit to implement a convex combination of channels
  • proof
  • Conjecture 1