This paper provides a new instance of quantum deletion error-correcting codes. This code can correct any single quantum deletion error, while our code is only of length 4. This paper also provides an example of an encoding quantum circuit and decoding quantum circuits. It is also proven that the length of any single deletion error-correcting codes is greater than or equal to 4. In other words, our code is optimal for the code length.
In this paper, we discuss a construction method of quantum deletion error-correcting codes. First of all, we define deletion errors for quantum states, an encoder, a decoder, and two conditions which is expressed by only the combinatorial language. Then, we prove that quantum deletion error-correcting codes can be constructed by two sets that satisfy the conditions. In other words, problems that correct the deletion errors for quantum states are reduced to problems that find the sets satisfying the condition by this paper. Also, we performed experiment of the codes over IBM Quantum Experience.
We propose two systematic constructions of deletion-correcting codes for protecting quantum information. The first one works with qudits of any dimension, but only one deletion is corrected and the constructed codes are asymptotically bad. The second one corrects multiple deletions and can construct asymptotically good codes. The second one also allows conversion of stabilizer-based quantum codes to deletion-correcting codes, and entanglement assistance.
This manuscript is an extended abstract version of the paper entitled ``Quantum Deletion Codes derived from Classical Deletion Codes.'' The paper contributes to the fundamental theory for quantum deletion error-correcting codes. The paper proposes a code construction condition for a partition of classical deletion error-correcting codes to derive quantum deletion error-correcting codes. The construction methods in this paper give examples of quantum codes that can correct single-quantum deletion errors and have a code rate arbitrarily close to 1, while the previously known quantum deletion cod…
We report new construction of nine-qubit error-correcting code, which introduces two new nine-qubit codes and one new three-qubit code. Because both the new two nine-qubit codes have the normal logical operators, as opposed to the nine-qubit Shor code, it results in different performance when the three codes are applied in concatenated quantum error-correction. On the other hand, one of the two nine-qubit codes has the same stabilizer generators as the nine-qubit Shor code, they are more suitable for the high-wight bit-flip noise, and the other code has the different stabilizer generators, whi…
We provide a systematic way of constructing entanglement-assisted quantum error-correcting codes via graph states in the scenario of preexisting perfectly protected qubits. It turns out that the preexisting entanglement can help beat the quantum Hamming bound and can enhance (not only behave as an assistance) the performance of the quantum error correction. Furthermore we generalize the error models to the case of not-so-perfectly-protected qubits and introduce the quantity infidelity as a figure of merit and show that our code outperforms also the ordinary quantum error-correcting codes.
A new class of error-correcting quantum codes is introduced capable of stabilizing qubits against spontaneous decay arising from couplings to statistically independent reservoirs. These quantum codes are based on the idea of using an embedded quantum code and exploiting the classical information available about which qubit has been affected by the environment. They are immediately relevant for quantum computation and information processing using arrays of trapped ions or nuclear spins. Interesting relations between these quantum codes and basic notions of design theory are established.
Suppose we are given two identical copies of an unknown quantum state and we wish to delete one copy from among the given two copies. The quantum no-deletion principle restricts us from perfectly deleting a copy but it does not prohibit us from deleting a copy approximately. Here we construct two types of a universal quantum deletion machine which approximately deletes a copy such that the fidelity of deletion does not depend on the input state. The two types of universal quantum deletion machines are (1) a conventional deletion machine described by one unitary operator and (2) a modified dele…
Large-scale universal quantum computing requires the implementation of quantum error correction (QEC). While the implementation of QEC has already been demonstrated for quantum memories, reliable quantum computing requires also the application of nontrivial logical gate operations to the encoded qubits. Here, we present examples of such operations by implementing, in addition to the identity operation, the NOT and the Hadamard gate to a logical qubit encoded in a five qubit system that allows correction of arbitrary single qubit errors. We perform quantum process tomography of the encoded gate…
We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b < 4; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n > 18. In ge…
We model repetitive quantum error correction (QEC) with the single-error-correcting five-qubit code on a network of individually-controlled qubits with always-on Ising couplings, using our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to serving as accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of …
We present an approach to one-way quantum computation (1WQC) that can compensate for single-qubit errors, by encoding the logical information residing on physical qubits into five-qubit error-correcting code states. A logical two-qubit cluster state that is the fundamental resource for encoded quantum teleportation is then described by a graph state containing ten vertices with constant degree seven. Universal 1WQC that incorporates error correction requires only multiple copies of this logical two-qubit state and a logical four-qubit linear cluster state, which are prepared only just in advan…
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.
We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization of the better-understood corruption error channels, and until recently, had mostly resisted effort toward a similar understanding as their corruption counterparts. They have received considerable attention in recent years. Very recently, Haeupler and Shahrasbi developed a framework, using what they call synchronisation strings, that allows one to turn insde…
We report the first nonadditive quantum error-correcting code, namely, a $((9,12,3))$ code which is a 12-dimensional subspace within a 9-qubit Hilbert space, that outperforms the optimal stabilizer code of the same length by encoding more levels while correcting arbitrary single-qubit errors.
A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors.
A universal set of quantum gates is constructed for the recently developed jump-error correcting quantum codes. These quantum codes are capable of correcting errors arising from the spontaneous decay of distinguishable qubits into statistically independent reservoirs. The proposed universal quantum gates are constructed with the help of Heisenberg- and Ising-type Hamiltonians acting on these physical qubits. This way it is guaranteed that the relevant error correcting code space is not left at any time even during the application of one of these quantum gates. The proposed entanglement gate is…
We derive a set of composite pulse sequences that generates CNOT gates and correct all systematic errors within the logical subspace to arbitrary order. These sequences are applicable for any two-qubit interaction Hamiltonian, and make no assumptions about the underlying noise mechanism except that it is constant on the timescale of the operation. We do assume access to error-free single-qubit gates, so single-qubit gate imperfections eventually limit the achievable fidelity. However, since single-qubit gates generally have much higher fidelities than two-qubit gates in practice, these pulse s…
Comparison of spin-qubit architectures for quantum error-correcting codes
Universidad de Costa Rica（哥斯达黎加大学）；RIKEN Center for Quantum Computing, RIKEN（理研量子计算中心，理研）；National Tsing Hua University（国立清华大学）；National Center for Excellence in Quantum Information Science and Engineering, National Tsing Hua University（国立清华大学量子信息科学与工程卓越国家中心）
Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where C_1 and C_2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C_1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubit…
Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this challenge, we experimentally realise the $[\![5,1,3]\!]$ code, the so-called smallest perfect code that permits c…
I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode $k=n-j-2$ qubits in $n=2^j$ qubits and correct $t=1$ error.
The recently introduced detected-jump correcting quantum codes are capable of stabilizing qubit-systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one detected jump-error correcting quantum codes is shown and the optimal redundancy, encoding and recovery as well as general properties of detected jump-error corre…
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use t…
Quantum error correction is a critical technique for transitioning from noisy intermediate-scale quantum (NISQ) devices to fully fledged quantum computers. The surface code, which has a high threshold error rate, is the leading quantum error correction code for two-dimensional grid architecture. So far, the repeated error correction capability of the surface code has not been realized experimentally. Here, we experimentally implement an error-correcting surface code, the distance-3 surface code which consists of 17 qubits, on the \textit{Zuchongzhi} 2.1 superconducting quantum processor. By ex…
It is shown that the noise process in quantum computation can be described by spatially correlated decoherence and dissipation. We demonstrate that the conventional quantum error correcting codes correcting for single-qubit errors are applicable for reducing spatially correlated noise.
Quantum error correction protocols have been developed to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behaviour of a quantum error-correcting code under general noise, including noisy measurements. This lack of knowledge is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop general methods for incorporating noisy measurement operations into simulations of quantum error-correcting codes and show that measurement errors on readout qubits manifest…
The corrected capacity of a quantum channel is defined as the best one-shot capacity that can be obtained by measuring the environment and using the result to correct the output of the channel. It is shown that (i) all qubit channels have corrected capacity log 2, (ii) a product of N qubit channels has corrected capacity N log 2, and (iii) all channels have corrected capacity at least log 2. The question is posed of finding the channel with smallest corrected capacity in any dimension d.
Universal quantum computers require a large network of qubits robust against errors. Recent theoretical and experimental studies on donor nuclear spins in silicon, engineered on semiconductor platforms compatible with industrial fabrication, show their coherent behavior and potential for scalability. Here we present a hardware-efficient quantum protocol that corrects phase flips of a nuclear spin using explicit experimentally feasible operations. We introduce the MAUS encoding (Moment AngUlar System encoding) which uses the large Hilbert space provided by the nuclear spin of the donor to encod…
We propose two quantum error correction schemes which increase the maximum storage time for qubits in a system of cold trapped ions, using a minimal number of ancillary qubits. Both schemes consider only the errors introduced by the decoherence due to spontaneous emission from the upper levels of the ions. Continuous monitoring of the ion fluorescence is used in conjunction with selective coherent feedback to eliminate these errors immediately following spontaneous emission events, and the conditional time evolution between quantum jumps is removed by symmetrizing the quantum codewords.
Given a ciphertext, is it possible to prove the deletion of the underlying plaintext? Since classical ciphertexts can be copied, clearly such a feat is impossible using classical information alone. In stark contrast to this, we show that quantum encodings enable certified deletion. More precisely, we show that it is possible to encrypt classical data into a quantum ciphertext such that the recipient of the ciphertext can produce a classical string which proves to the originator that the recipient has relinquished any chance of recovering the plaintext should the decryption key be revealed. Our…
Although Majorana platforms are promising avenues to realizing topological quantum computing, they are still susceptible to errors from thermal noise and other sources. We show that the error rate of Majorana qubits can be drastically reduced using a 1D repetition code. The success of the code is due the imbalance between the phase error rate and the flip error rate. We demonstrate how a repetition code can be naturally constructed from segments of Majorana nanowires. We find the optimal lifetime may be extended from a millisecond to over one second.
In multi-qubit system, correlated errors subject to unwanted interactions with other qubits is one of the major obstacles for scaling up quantum computers to be applicable. We present two approaches to correct such noise and demonstrate with high fidelity and robustness. We use spectator and intruder to discriminate the environment interacting with target qubit in different parameter regime. Our proposed approaches combines analytical theory and numerical optimization, and are general to obtain smooth control pulses for various qubit systems. Both theory and numerical simulations demonstrate t…
Low-frequency noise presents a serious source of decoherence in solid-state qubits. When combined with a continuous weak measurement of the eigenstates, the low-frequency noise induces a second-order relaxation between the qubit states. Here we show that the relaxation provides a unique approach to calibrate the low-frequency noise in the time-domain. By encoding one qubit with two physical qubits that are alternatively calibrated, quantum logic gates with high fidelity can be performed.
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 {\em any} permutation-invariant quantum code that has a distance of $t+1$ can correct $t$ quantum deletions for any positive integer $t$ in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corr…
The noise in physical qubits is fundamentally asymmetric: in most devices, phase errors are much more probable than bit flips. We propose a quantum error correcting code which takes advantage of this asymmetry and shows good performance at a relatively small cost in redundancy, requiring less than a doubling of the number of physical qubits for error correction.
AI-Enabled Decoding of Qubit Loss for Quantum Error-Correcting Codes
Tsinghua University（清华大学）；Intelligent Quantum Inception Co., Ltd.（智量量子创新有限公司）；Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；iFLYTEK Research（科大讯飞研究院）
This paper presents conditions for constructing permutation-invariant quantum codes for deletion errors and provides a method for constructing them. Our codes give the first example of quantum codes that can correct two or more deletion errors. Also, our codes give the first example of quantum codes that can correct both multiple-qubit errors and multiple-deletion errors. We also discuss a generalization of the construction of our codes at the end.
We analyze the continuous operation of the nine-qubit error correcting Bacon-Shor code with all noncommuting gauge operators measured at the same time. The error syndromes are continuously monitored using cross-correlations of sets of three measurement signals. We calculate the logical error rates due to $X$, $Y$ and $Z$ errors in the physical qubits and compare the continuous implementation with the discrete operation of the code. We find that both modes of operation exhibit similar performances when the measurement strength from continuous measurements is sufficiently strong. We also estimat…
Most quantum computing architectures can be realized as two-dimensional lattices of qubits that interact with each other. We take transmon qubits and transmission line resonators as promising candidates for qubits and couplers; we use them as basic building elements of a quantum code. We then propose a simple framework to determine the optimal experimental layout to realize quantum codes. We show that this engineering optimization problem can be reduced to the solution of standard binary linear programs. While solving such programs is a NP-hard problem, we propose a way to find scalable optima…
We study the encoding complexity for quantum error correcting codes with large rate and distance. We prove that random Clifford circuits with $O(n \log^2 n)$ gates can be used to encode $k$ qubits in $n$ qubits with a distance $d$ provided $\frac{k}{n} < 1 - \frac{d}{n} \log_2 3 - h(\frac{d}{n})$. In addition, we prove that such circuits typically have a depth of $O( \log^3 n)$.
