Finding good quantum codes using the Cartan form

Jayashankar, Akshaya; Babu, Anjala M; Ng, Hui Khoon; Mandayam, Prabha

doi:10.1103/physreva.101.042307

Cited by 8 publications

(8 citation statements)

References 35 publications

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“…Recently, a systematic way of constructing optimal noise-adapted approximate codes for general noise models using the Cartan decomposition was provided in Ref. [25]. Furthermore, in contrast to earlier numerical approaches, the worst-case fidelity was used as the figure of merit to obtain the optimal codes.…”

Section: Noise-adapted Quantum Codesmentioning

confidence: 99%

“…10. Specifically, for a qubit noise channel E and two-dimensional code C, using the Petz map leads to a simple optimization problem for the worst-case fidelity, namely [25],…”

Section: Noise-adapted Quantum Codesmentioning

confidence: 99%

“…We will first review the early theoretical progress on approximate QEC [19][20][21][22], and introduce an important noiseadapted recovery map, namely, the Petz map [23,24]. We then discuss various recent approaches to constructing noise-adapted quantum codes [25][26][27][28]. Next we survey some interesting applications of noise-adapted recovery maps, especially in the context of many-body quantum systems.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Error Correction: Noise-Adapted Techniques and Applications

Jayashankar

Mandayam²

2022

J Indian Inst Sci

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The quantum computing devices of today have tens to hundreds of qubits that are highly susceptible to noise due to unwanted interactions with their environment. The theory of quantum error correction provides a scheme by which the effects of such noise on quantum states can be mitigated, paving the way for realising robust, scalable quantum computers. In this article we survey the current landscape of quantum error correcting (QEC) codes, focusing on recent theoretical advances in the domain of noise-adapted QEC, and highlighting some key open questions. We also discuss the interesting connections that have emerged between such adaptive QEC techniques and fundamental physics, especially in the areas of many-body physics and cosmology. We conclude with a brief review of the theory of quantum fault tolerance which gives a quantitative estimate of the physical noise threshold below which error-resilient quantum computation is possible.

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Section: Noise-adapted Quantum Codesmentioning

confidence: 99%

“…10. Specifically, for a qubit noise channel E and two-dimensional code C, using the Petz map leads to a simple optimization problem for the worst-case fidelity, namely [25],…”

Section: Noise-adapted Quantum Codesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum Error Correction: Noise-Adapted Techniques and Applications

Jayashankar

Mandayam²

2022

J Indian Inst Sci

Self Cite

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“…Refs. [16,[24][25][26] designed classical algorithms for finding quantum codes associated with graphs. Ref.…”

Section: Introductionmentioning

confidence: 99%

Quantum variational learning for quantum error-correcting codes

Cao

Zhang

et al. 2022

Quantum

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Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., ((n,2n−6,3))2 for n from 7 to 14. We also found new ((6,2,3))2 and ((7,2,3))2 codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a ((7,3,3))2 code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.

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“…First, the gradient of Eq. ( 6) can be obtained analytically; second, this allows us to apply K on v exactly (see supplementary material); and third, since a circuit implementation for individual Pauli strings is known [18,19], we avoid the need for further decomposition of K.…”

mentioning

confidence: 99%

Fixed Depth Hamiltonian Simulation via Cartan Decomposition

Kökcü,

Steckmann,

Wang

et al. 2021

Preprint

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Simulating spin systems on classical computers is challenging for large systems due to the significant memory requirements. This makes Hamiltonian simulation by quantum computers a promising option due to the direct representation of quantum states in terms of its qubits. Standard algorithms for time evolution on quantum computers require circuits whose depth grows with time. We present a new algorithm, based on Cartan decomposition of the Lie algebra generated by the Hamiltonian, that generates a circuit with time complexity O(1) for ordered and disordered models of n spins. We highlight our algorithm for special classes of models where an O(n 2 )-gate circuit emerges. Compared to product formulas with significantly larger gate counts, our algorithm drastically improves simulation precision. Our algebraic technique sheds light on quantum algorithms and will reduce gate requirements for near term simulation.

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Finding good quantum codes using the Cartan form

Cited by 8 publications

References 35 publications

Quantum Error Correction: Noise-Adapted Techniques and Applications

Quantum Error Correction: Noise-Adapted Techniques and Applications

Quantum variational learning for quantum error-correcting codes

Fixed Depth Hamiltonian Simulation via Cartan Decomposition

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