Quantum variational learning for quantum error-correcting codes

Abstract

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.

Quantum Error Correction with Quantum Autoencoders

“…To allow for more flexible denoising strategies it would be desirable to search for entirely new logical encodings that are optimally suited to protect quantum information from unknown types of noise. Some schemes have already been proposed on how quantum neural networks can be used to achieve this [66][67][68].…”

“…Compression of quantum data using QAEs has already been achieved in experiments using single photons [63,64] or superconducting qubits [65]. In other works, certain types of quantum neural networks were proposed to find suitable encodings of quantum information into logical states that allow for hardware-specific noise to be corrected [66][67][68].…”

Mitigating algorithmic errors in quantum optimization through energy extrapolation

“…Denote the energy and variance data sets as S E and S var . After several samples of the initial values and optimization, we find the smallest energy min(S E ) and the smallest variance min(S var ), keep only the energy-variance data that satisfy E < min(S E ) + R E and Δ var < min(S var ) + R var , (23) where R var are R E are small intervals that in principle should scale with the system size. Here, we choose R var = min(S var ), R E = 0.5.…”

“…Another class of NISQ algorithms that has attracted particular attention in recent years are variational quantum algorithms [7][8][9][14][15][16][17][18][19][20][21][22][23]. Unlike QA, these algorithms also require a classical computer to carry out an optimization that complements the quantum processing.…”

Quantum Coding via Quasi-Cyclic Block Matrix

“…Furthermore, due to some fields in quantum information becoming gradually more practical, this has prompted researchers to identify a good coding method for quantum error-correcting codes of long length. For example, encoding large-scale data has potential applications in the field of machine learning (ML) with respect to big data [ 11 , 12 , 13 , 14 ]. Therefore, how to efficiently express classical massive data with physics-based codes is also an important research field.…”