Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition.

Chapter 7 - Quantum error correction fundamentals

We analyze simple quantum error detection and quantum error correction protocols relevant to current experiments with superconducting qubits. We show that for qubits with energy relaxation the repetitive $N$-qubit codes cannot be used for quantum error correction, but can be used for quantum error detection. In the latter case it is sufficient to use only two qubits for the encoding. In the analysis we demonstrate a useful technique of unraveling the qubit energy relaxation into ``relaxation'' and ``no relaxation'' scenarios. Also, we propose and numerically analyze several two-qubit algorithms for quantum error detection and correction, which can be readily realized at the present-day level of the phase qubit technology.

The storage and processing of quantum information are susceptible to external noise, resulting in computational errors. A powerful method to suppress these effects is quantum error correction. Typically, quantum error correction is executed in discrete rounds, using entangling gates and projective measurement on ancillary qubits to complete each round of error correction. Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancillary qubits, and their associated errors. An FPGA controller actively corrects errors as they are detected, achieving an average bit-flip detection efficiency of up to 91%. Furthermore, the protocol increases the relaxation time of the protected logical qubit by a factor of 2.7 over the relaxation times of the bare comprising qubits. Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.

Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.

Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.

Chapter 7 - Quantum Error Correction

Quantum error correction (QEC) can overcome the errors experienced by qubits and is therefore an essential component of a future quantum computer. To implement QEC, a qubit is redundantly encoded in a higher-dimensional space using quantum states with carefully tailored symmetry properties. Projective measurements of these parity-type observables provide error syndrome information, with which errors can be corrected via simple operations. The 'break-even' point of QEC--at which the lifetime of a qubit exceeds the lifetime of the constituents of the system--has so far remained out of reach. Although previous works have demonstrated elements of QEC, they primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to preserve a qubit over time. Here we demonstrate a QEC system that reaches the break-even point by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of Schrödinger-cat states of a superconducting resonator. We implement a full QEC protocol by using real-time feedback to encode, monitor naturally occurring errors, decode and correct. As measured by full process tomography, without any post-selection, the corrected qubit lifetime is 320 microseconds, which is longer than the lifetime of any of the parts of the system: 20 times longer than the lifetime of the transmon, about 2.2 times longer than the lifetime of an uncorrected logical encoding and about 1.1 longer than the lifetime of the best physical qubit (the |0〉f and |1〉f Fock states of the resonator). Our results illustrate the benefit of using hardware-efficient qubit encodings rather than traditional QEC schemes. Furthermore, they advance the field of experimental error correction from confirming basic concepts to exploring the metrics that drive system performance and the challenges in realizing a fault-tolerant system.

Pre-shared non-local entanglement dramatically simplifies and improves the performance of quantum error correction via entanglement-assisted quantum error-correcting codes (EAQECCs). However, even considering the noise in quantum communication only, the non-local sharing of a perfectly entangled pair is technically impossible unless additional resources are consumed, such as entanglement distillation, which actually compromises the efficiency of the codes. Here we propose an error-correcting protocol assisted by two-way noisy communication that is more easily realisable: all quantum communication is subjected to general noise and all entanglement is created locally without additional resources consumed. In our protocol the pre-shared noisy entangled pairs are purified simultaneously by the decoding process. For demonstration, we first present an easier implementation of the well-known EAQECC [[4, 1, 3; 1]]. Then, we construct for the first time a 1-error-correcting code of 4 physical qudits capable of encoding 1 qubit of information, which is impossible in standard quantum error correction, demonstrating that our protocol can also improve the encoding rate in some sense. A systematic construction of two-way-noisy-communication-assisted codes is also provided.

Quantum machine learning (QML) is an emerging field that promises advantages such as faster training, improved reliability and superior feature extraction over classical counterparts. However, its implementation on quantum hardware is challenging due to the noise inherent in these systems, necessitating the use of quantum error correction (QEC) codes. Current QML research remains primarily theoretical, often assuming noise-free environments and offering little insight into the integration of QEC with QML implementations. To address this, we investigate the performance of a simple, parity-classifying Variational Quantum Classifier (VQC) implemented with the [[4,2,2]] error-detecting stabiliser code in a simulated noisy environment. To our knowledge, this is the first implementation of a stabiliser-based error detection code to a QML algorithm. We invoke ancilla qubits to logically encode rotation gates, and classically simulate the logically-encoded VQC under two simple noise models representing gate noise and environmental noise. We demonstrate that the stabiliser code improves the training accuracy at convergence compared to noisy implementations without error detection. However, we find that the effectiveness and reliability of error detection is contingent upon keeping the ancilla qubit error rates below a specific threshold, due to the propagation of ancilla errors to the physical qubits. Our results provide an important insight that extends to QML implemented with full error-correction: when the QEC code requires ancilla qubits for logical rotations but cannot fully correct errors propagated between ancilla and physical qubits, the maximum achievable accuracy of the QML model is constrained. This highlights the need for additional error correction or mitigation strategies to support the practical implementation of QML algorithms with QEC on quantum devices.

Quantum error correction is a set of methods to protect quantum information—that is, quantum states—from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely used codes are the stabilizer codes, which are closely related to classical linear codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators; the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[n,k,d]], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n>k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design. Other approaches to error mitigation in quantum systems include decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.

Noisy intermediate scale quantum (NISQ) systems are susceptible to errors that culminate in near-one hundred percent data loss. This is due to quantum state fragility and the incredibly high quantum communication error rates caused by decoherence, or quantum noise. As such, stabilizing qubit operational imprecision in quantum information processing is a critical area of research in quantum computing. Adaptive quantum machine learning (QML) methods, like unsupervised and fully entangled quantum generative adversarial networks is one such technology theorized to provide a breakthrough in quantum error suppression. Mechanizing the quantum error detection and correction process with QML provides a path forward from today’s monolithic quantum computers running almost exclusively single-core quantum processing unit (QPU) designs, to the next generation of federated quantum computers using multi-core QPUs. Automating the detection and correction of quantum errors in powerful NISQ devices will pave the way for fault-tolerant quantum computing, making quantum speeds at quantum scale suddenly achievable.

Scalable quantum computation and communication require error control to protect quantum information against unavoidable noise. Quantum error correction protects information stored in two-level quantum systems (qubits) by rectifying errors with operations conditioned on the measurement outcomes. Error-correction protocols have been implemented in nuclear magnetic resonance experiments, but the inherent limitations of this technique prevent its application to quantum information processing. Here we experimentally demonstrate quantum error correction using three beryllium atomic-ion qubits confined to a linear, multi-zone trap. An encoded one-qubit state is protected against spin-flip errors by means of a three-qubit quantum error-correcting code. A primary ion qubit is prepared in an initial state, which is then encoded into an entangled state of three physical qubits (the primary and two ancilla qubits). Errors are induced simultaneously in all qubits at various rates. The encoded state is decoded back to the primary ion one-qubit state, making error information available on the ancilla ions, which are separated from the primary ion and measured. Finally, the primary qubit state is corrected on the basis of the ancillae measurement outcome. We verify error correction by comparing the corrected final state to the uncorrected state and to the initial state. In principle, the approach enables a quantum state to be maintained by means of repeated error correction, an important step towards scalable fault-tolerant quantum computation using trapped ions.

<p indent="0mm">Recently, the achievement, entitled “lifetime enhancement for a logic qubit by bosonic encoding error correction”, has been selected as one of “The Ten Major Advances in Chinese Science 2023”. Here, we will introduce briefly the importance of this work. It is known that quantum states are fragile due to decoherence induced by environment. Error will happen on a qubit, resulting in continuously changes of the parameters for this qubit, such as amplitude and phase parameters. The aim of quantum error correction is to amend those errors by using various quantum error correction codes, in which a logic qubit can be represented by several physical qubits. It should be noted that the quantum error correction can be realized by correcting a discrete set of errors, such as bit-flip error, phase-flip error or both of them. On the other hand, types of errors occurred depend on specific platforms of quantum computation. For example, photon loss is more inclined to occur in photonic systems. In this case, the approach of bosonic encoding of quantum error correction is more efficient. The work “lifetime enhancement for a logic qubit by bosonic encoding error correction” is for such a scenario. The challenge of quantum error correction is that operation itself is not perfect which may incur new errors. So it is actually difficult to demonstrate that we can benefit from quantum error correction. The achievement of lifetime enhancement by bosonic encoding error correction succeeds in beating this break-even point. The experiments are performed by circuit quantum electrodynamics on a superconducting processor with a qubit and a coupled resonate cavity. The logic qubit is realized by the bosonic mode of the cavity, while the superconducting qubit plays the role of the ancillary qubit in error correction. The initial states are prepared by the superconducting qubit, and are transferred to the logic qubit in the cavity. Then the error correction can be performed based on detection of photon loss by the superconducting qubit. The results show that the lifetime of the logic qubit increases from <sc>694 μs</sc> to <sc>805 μs</sc> with 16% enhancement. Besides high quality devices, the success of this experiment depends on the feedback control to detect and correct the error of photon loss in a short time. The result of beating the break-even point of bosonic encoding error correction is for one logic qubit. In the near future, it is hopeful to implement logic gates, such as single-qubit rotation gate and two-qubit gate, on a couple of logic qubits by bosonic encoding. The importance of the results will depend on whether higher fidelity can be achieved compared with those of gates with physical qubits. In this way, the advantage of the logic qubits for quantum computation is presented. It will be significant that the approach of bosonic encoding with a superconducting processor is scalable to hundreds of logic qubits, which may take a few years to realize. Our aim is to realize a fault-tolerant universal quantum computer based on a large quantity of high-precision logic qubits. The work “lifetime enhancement for a logic qubit by bosonic encoding error correction” is an important step toward this aim.

In this work, we investigate some applications of quantum information theory motivated by high-energy physics. There is strong evidence suggesting that entanglement is deeply connected with the geometry of spacetime, which leads to surprising applications of quantum information theory in the AdS/CFT correspondence and holography. We start by reviewing the fundamental concepts of the AdS/CFT correspondence which play a key role in bulk-boundary reconstruction, in particular, we explore some features which suggest that one must interpret the encoding of information in the correspondence as a quantum errorcorrecting code. We discuss the fundamentals of error correction, exploring the formalisms of operator algebra and stabilizer codes. Then, we establish the concrete connection between the two main concepts by showing examples of quantum error-correcting codes that serve as a toy model for AdS/CFT. We illustrate how the 3-qutrit code and the HaPPY code can be powerful tools to explore the correspondence analytically and to solve apparent paradoxes. Following recent results, using quantum error correction, that suggest an intrinsic incompatibility of quantum gravity with global symmetries, we explore approximate error-correcting codes and asymmetric codes as a way to better understand the consequences in a quantum resource-theoretic way. Finally, we discuss our original contribution: geometric bounds for approximate quantum error correction. We calculate our bounds for three typical quantum channels that model the lack of exactness in error correction, namely, dephasing, depolarizing, and amplitude damping channels. The implications of our bounds for AdS/CFT are somewhat elusive; nonetheless, we provide a new approach to benchmark approximations in error correction performance, which may be of high interest for AdS/CFT and its corresponding absence of global symmetries.

Quantum communication offers transformative potential for media transmission by addressing the limitations of classical communication systems. To realize this potential, the study proposes a quantum communication framework for transmitting compressed videos over error-prone channels, leveraging quantum superposition. Two channel coding schemes are analyzed: quantum error correction (three-qubit, five-qubit, and seven-qubit codes) and classical error correction (1/3 rate polar code), all operating within the same bandwidth constraints. The proposed systems are benchmarked against a classical communication system using 1/3 rate polar codes. Results show that the three-qubit error correction-based quantum communication system, while simple and efficient, achieves significant performance gains over both classical error correction-based quantum and classical communication systems, with up to 41.42 dB in peak signal-to-noise ratio (PSNR), 0.9639 in structural similarity index measure (SSIM), and 94.4042 in video multimethod assessment fusion (VMAF). However, the five-qubit and seven-qubit systems outperform the three-qubit system, with the seven-qubit system surpassing all others in high noise environments, demonstrating its robustness across various group of pictures (GOP) formats. These findings highlight the trade-offs between simplicity and complexity, as the three-qubit system is practical and efficient, while the five-qubit and seven-qubit channel codes offer higher fidelity and resilience at the cost of increased complexity.