Optimal universal quantum error correction via bounded reference frames

Open Access

Optimal universal quantum error correction via bounded reference frames

Yuxiang Yang, Yin Mo, Joseph M. Renes, Giulio Chiribella, and Mischa P. Woods

Phys. Rev. Research 4, 023107 – Published 9 May 2022

Abstract

Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to the entire group of local unitary gates in dimension $d (< \infty)$ , using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the $n$ qudits upon which the code is built are erased, our covariant code has an error that scales as $1 / n^{2}$ , which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as $1 / n$ . We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.

Received 21 January 2021
Revised 11 June 2021
Accepted 5 April 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.023107

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum error correction Quantum metrology

Quantum Information, Science & Technology

Authors & Affiliations

Yuxiang Yang ^1,2,*, Yin Mo², Joseph M. Renes ¹, Giulio Chiribella^2,3,4,5, and Mischa P. Woods¹

¹Institute for Theoretical Physics, ETH Zürich, Switzerland
²QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
³Department of Computer Science, Parks Road, Oxford OX1 3QD, United Kingdom
⁴Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
⁵The University of Hong Kong Shenzhen Institute of Research and Innovation, 5/F, Key Laboratory Platform Building, No.6, Yuexing 2nd Rd., Nanshan, Shenzhen 518057, China

^*mischa@phys.ethz.ch

Article Text

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Issue

Vol. 4, Iss. 2 — May - July 2022

Subject Areas

Quantum Physics

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Images

Figure 1
Error correction via reference frames. The physical space of the computer is over $n$ qudits which are divided into two subspaces: the computational space (black and green) and a reference frame space (yellow and brown). The former is where the gates needed for the algorithm are applied transversally, while the latter is where the reference frames are located. The two systems are only classically correlated at any given moment in our protocols.
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Figure 2
Covariant encoder construction. (The following is an in-principle method to implement our QECC. For practical implementations, see section Towards fault-tolerant QECCs.) Given an arbitrary channel $E$ encoding one logical qudit into $n_{C}$ qudits, and a corresponding decoder $D$ , our covariant encoding protocol runs as follows. (a) Choose a unitary $U \in SU (d)$ at random (in practice, this can be achieved by sampling from a large enough finite set of gates) and apply its inverse unitary channel $U_{L}^{- 1}$ to the logical state $ρ_{L}$ one wishes to encode. (b) Perform $U_{C} \circ E$ , where $U_{C}$ is the unitary channel corresponding to the representation of $U$ in the computational space, and $U_{C}$ are local gates which will be transversal for the new code. (c) Such an operation alone does not result in a useful code, as the identification of the particular $U$ is needed to later correct errors and decode. Thus, to store $U$ efficiently, we append $n_{R}$ quantum reference frames, and (d) apply it to a quantum reference frame, resulting in the unitary's identity being encoded into the reference frame's state. The outcome of steps 1 to 4 is a new code with a covariant encoder $E_{cov}$ which outputs an encoded state on the original computational and reference frame registers, C and R.
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Figure 3
Weak erasure error with “five-qubit code”. Under the weak erasure model, the decoding error of our protocol (with the five-qubit code [49] as the subroutine noncovariant code and the generalized sine states as the reference frame) is plotted as a function of $2 m$ , the number of qubits contained in each block of the reference frame register. The blue dots correspond to the evaluated values of the decoding error and the red line corresponds to the fitting curve $\frac{269.2}{{(2 m)}^{2}}$ .
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Figure 4
i.i.d. dephasing/depolarizing with $p = 0.2$ and “five-qubit code”. The decoding error for i.i.d. dephasing/depolarizing error of the our covariant code using “five-qubit code” as the subroutine encoding and $n_{R} / 2$ maximally entangled states as the reference frame is plotted as a function of $n$ , the number of total physical qubits. The blue dots correspond to the evaluated values of the decoding error and the red lines correspond to the fitting curves: $32.43 / n$ for the dephasing error and $33.30 / n$ for the depolarizing error.
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Figure 5
Reference frame assisted covariant error correction. An arbitrary code $(E, D)$ can be converted into a covariant code $(E_{cov}, D_{cov})$ following the instruction of Protocol 2. The subroutine encoder $E : H_{L} \to H_{C}$ is made covariant via twirling, where $U$ is a random unitary sampled from the Haar measure. A reference frame register, which is divided into multiple copies of a reference frame state $ψ$ , is employed to keep track of the unitary. Since $E_{cov}$ is covariant, any desired logical operation $V$ can be implemented transversally. To correct error (characterized by a channel $C$ ) or to decode the information, the reference frame register is first measured, and recovery operations are performed on the computational register depending on the measurement outcome.
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