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Reconfiguring Quantum Error-correcting Codes for Real-life Errors

Muhammad Ahsan

Published Jul 30, 2023• 1 citation
Publisher

Abstract

How to experimentally show that a quantum gate protected by a fully error-corrected code (e.g. [[7,1,3]] has higher fidelity than that of its physical counterpart, without encoded state-preparation. The experimental results are based on IBMQ device ibmq_melbourne.
DOI
10.36227/techrxiv.23786790.v1

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Abstract High error-rates preclude the preparation of fully error-corrected logical qubit state on noisy intermediate scale quantum (NISQ) computers. When operand logical qubits inherit large state-preparation noise, it is difficult to show that subsequent logical gate fails less frequently than its physical (unprotected) version. We articulate a scheme of decoupling transversal logical gate errors from state-preparation noise and experimentally validate its use-case for IBM Q quantum processors. We find that in the absence of state preparation noise, the IBM Q processors significantly raise the likelihood of certain two-qubit errors in the operand(s) of [[7, 1, 3]] transversal gates. Yet, encoding can still be shown to improve the gate fidelity provided that the gate operands are strategically decoded/corrected for the likely two-qubit errors in lieu of their less likely single-qubit counterparts. This trade-off enables quantum CSS code to principally correct longer strings of errors without increasing the codeword size and paves new avenues of investigating fault-tolerance in NISQ computers.
Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here, we use a trapped-ion system to experimentally prepare mmm-qubit Greenberger-Horne-Zeilinger states and sample the measurement results to construct $m\ifmmode\times\else\texttimes\fi{}m$ logical states of the [[m2,1,m]][[{m}^{2},1,m]][[m2,1,m]] Shor code, up to m=7m=7m=7. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the optimal code size depends on the physical error rate and we find that m=5m=5m=5 has the best performance in our system. We further realize the direct logical encoding of the [[9,1,3]][[9,1,3]][[9,1,3]] Shor code on nine qubits in a 13-ion chain for comparison, with $98.8(1)\mathrm{%}$ and $98.5(1)\mathrm{%}$ fidelity for state ${|\ifmmode\pm\else\textpm\fi{}⟩}_{L}$, respectively.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.