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Physical Review A

Initializing a permutation-invariant quantum error-correction code

Chunfeng Wu, Yimin Wang, Chu Guo, Yingkai Ouyang, Gangcheng Wang, Xun‐Li Feng

Published Jan 22, 2019• 13 citations
Publisher

Abstract

Recently, there has been growing interest in using quantum error correction in practical devices. A central issue in quantum error correction is the initialization of quantum data into a quantum error-correction code. Most studies have concentrated on generating quantum codes based on their encoding quantum circuits. However, this often leads to a large number of steps required in the initialization, and hence this process can be prone to errors. The purpose of this work is to demonstrate that permutation-invariant quantum error-correction codes can be created with high fidelity by exploiting their underlying symmetry. The code is initialized on multiple qubits that mutually interact or are themselves coupled to a quantum harmonic oscillator. We show that the so-called selective resonant interaction is derivable on such physical systems. By utilizing the selective resonant interaction, these highly symmetric codes may be rapidly generated with excellent fidelity. We also discuss the potential of initializing permutation-invariant quantum error-correction codes based on the state-of-art experimental techniques.

DOI
10.1103/physreva.99.012335

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Recently, there has been growing interest in using quantum error correction in practical devices. A central issue in quantum error correction is the initialization of quantum data into a quantum error-correction code. Most studies have concentrated on generating quantum codes based on their encoding quantum circuits. However, this often leads to a large number of steps required in the initialization, and hence this process can be prone to errors. The purpose of this work is to demonstrate that permutation-invariant quantum error-correction codes can be created with high fidelity by exploiting their underlying symmetry. The code is initialized on multiple qubits that mutually interact or are themselves coupled to a quantum harmonic oscillator. We show that the so-called selective resonant interaction is derivable on such physical systems. By utilizing the selective resonant interaction, these highly symmetric codes may be rapidly generated with excellent fidelity. We also discuss the potential of initializing permutation-invariant quantum error-correction codes based on the state-of-art experimental techniques.
The increasing interest in using quantum error correcting codes in practical devices has heightened the need for designing quantum error correcting codes that can correct against specialized errors, such as that of amplitude damping errors which model photon loss. Although considerable research has been devoted to quantum error correcting codes for amplitude damping, not so much attention has been paid to having these codes simultaneously lie within the decoherence free subspace of their underlying physical system. One common physical system comprises of quantum harmonic oscillators, and constant-excitation quantum codes can be naturally stabilized within them. The purpose of this paper is to give constant-excitation quantum codes that not only correct amplitude damping errors, but are also immune against permutations of their underlying modes. To construct such quantum codes, we use the nullspace of a specially constructed matrix based on integer partitions.
A permutation-invariant code on mmm qubits is a subspace of the symmetric subspace of the mmm qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading-order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation-invariant codes and quantum error correction.
Quantum synchronisation errors are a class of errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are ttt-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are (t,s)(t,s)(t,s)-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.
Quantum Error Recovery (QER) uses knowledge of the error channel acting on a quantum system to find optimal recovery maps. The scheme restores the uncorrupted state with a fidelity exceeding that achieved by noise parameter independent quantum error correction. We use a generic coherent QER map implemented with a quantum circuit acting on the system together with ancillary qubits to recover quantum information stored in permutation invariant (PI) codes. PI codes admit tunable parameters to suit the noise model and benefit from simple recovery operation circuits with reduced addressability requirements, unlike stabilizer codes. We showcase the method by modeling QER in PI codes after collective and local symmetric correlated amplitude-damping (AD) noise, a non-Pauli noise process for which stabilizer codes often require additional overhead. We also propose a new PI code family called CAD codes with explicit examples on 4 and 9 qubits for global symmetric AD errors. We show that CAD9 (supported on 9 qubits) code beats many existing codes by more than one order of magnitude. For the CAD4 code, which perfectly corrects 1 global symmetric AD error, the compiled recovery circuit consists of 10 system and system-ancilla gates which can be realized from linear geometric phase gates. Our work provides a direct path from optimized recovery maps to experimentally implementable, low-overhead protocols.
A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to t2{t}^{2}t2; one family of codes perfectly corrects arbitrary weight ttt errors, while the other family of codes approximately correct ttt spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.
Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that any permutation-invariant quantum code that has a distance oft+1t+1t+1can correcttttquantum deletions for any positive integertttin both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corresponding coding bounds for permutation-invariant quantum codes under quantum deletions. We focus our attention on a specific family ofNNN-qubit permutation-invariant quantum codes, which we call shifted gnu codes. The main result of this work is that their encoding and decoding algorithms can be performed inO(N)O(N)O(N)andO(N2)O(N^{2})O(N2).