Five-qubit Code Research Articles

Fault-tolerant double-circular connectivity pattern for quantum stabilizer codes

Recently, the circular connectivity pattern has been presented for a class of stabilizer quantum error correction codes. The circular connectivity pattern for such a class of stabilizer codes can be implemented in a resource-efficient manner using a single ancilla and native two-qubit Controlled-Not-Swap gates (CNS) gates, which may be interesting for demonstrating error-correction codes with superconducting quantum processors. However, one concern is that this scheme is not fault-tolerant. And it might not apply to the Calderbank-Shor-Steane (CSS) codes. In this paper, we present a fault-tolerant version of the circular connectivity pattern, named the double-circular connectivity pattern. This pattern is an implementation for syndrome-measurement circuits with a flagged error correction scheme for stabilizer codes. We illustrate that this pattern is available for Steane code (a CSS code), Laflamme’s five-qubit code, and Shor’s nine-qubit code. For Laflamme’s five-qubit code and Shor’s nine-qubit code, the pattern has the property that it uses only native two-qubit CNS gates, which are more efficient in the superconducting quantum platform.

Exact performance of the five-qubit code with coherent errors

Exact performance of the five-qubit code with coherent errors

Tailored XZZX codes for biased noise

Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.

Fault-tolerant operation of a logical qubit in a diamond quantum processor

Solid-state spin qubits is a promising platform for quantum computation and quantum networks1,2. Recent experiments have demonstrated high-quality control over multi-qubit systems3–8, elementary quantum algorithms8–11 and non-fault-tolerant error correction12–14. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations15–18. Overcoming imperfections in this way remains an important outstanding challenge for quantum science15,19–27. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits28–30. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.

Device-Independent Certification of Genuinely Entangled Subspaces.

Self-testing is a procedure for characterizing quantum resources with the minimal level of trust. Up to now it has been used as a device-independent certification tool for particular quantum measurements, channels, and pure entangled states. In this work we introduce the concept of self-testing more general entanglement structures. More precisely, we present the first self-tests of an entangled subspace-the five-qubit code and the toric code. We show that all quantum states maximally violating a suitably chosen Bell inequality must belong to the corresponding code subspace, which remarkably includes also mixed states.

Pieceable fault tolerant conversion between 5-qubit code and 7-CSS code

We propose a non-transversal but pieceable fault tolerant conversion circuit that is used to convert encoded information between five-qubit code and seven-qubit CSS code. Since a syndrome extraction circuit requiring fewer ancillary qubit resources would facilitate the realization of large-scale quantum computations, we further adapt a flag-assisted fault tolerant syndrome measurement scheme to reduce the cost of ancillary preparation. Numerical simulations are also performed to further analyze the performance of our conversion method.

An integrity measure to benchmark quantum error correcting memories

Rapidly developing experiments across multiple platforms now aim to realise small quantum codes, and so demonstrate a memory within which a logical qubit can be protected from noise. There is a need to benchmark the achievements in these diverse systems, and to compare the inherent power of the codes they rely upon. We describe a recently introduced performance measure called integrity, which relates to the probability that an ideal agent will successfully ‘guess’ the state of a logical qubit after a period of storage in the memory. Integrity is straightforward to evaluate experimentally without state tomography and it can be related to various established metrics such as the logical fidelity and the pseudo-threshold. We offer a set of experimental milestones that are steps towards demonstrating unconditionally superior encoded memories. Using intensive numerical simulations we compare memories based on the five-qubit code, the seven-qubit Steane code, and a nine-qubit code which is the smallest instance of a surface code; we assess both the simple and fault-tolerant implementations of each. While the ‘best’ code upon which to base a memory does vary according to the nature and severity of the noise, nevertheless certain trends emerge.

Universal set of dynamically protected gates for bipartite qubit networks: Soft pulse implementation of the [[5,1,3]] quantum error-correcting code

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. We use our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to being accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating the unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of decoherence. The combined DD and QEC protocol dramatically improves the coherence, with the QEC alone being responsible for more than an order of magnitude infidelity reduction.

Generality of the concatenated five-qubit code

In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used to construct a near-perfect effective qubit channel (with a error below $10^{-5}$) from arbitrary noise channels. The exact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicit protocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary size of the concatenated five-qubit code depends only on the entanglement fidelity of the initial noise channels.

Trapped-ion quantum error-correcting protocols using only global operations

Quantum error-correcting codes are many-body entangled states that are prepared and measured using complex sequences of entangling operations. Each element of such an entangling sequence introduces noise to delicate quantum information during the encoding or reading out of the code. It is important therefore to find efficient entangling protocols to avoid the loss of information. Here we propose an experiment that uses only global entangling operations to encode an arbitrary logical qubit to either the five-qubit repetition code or the five-qubit code, with a six-ion Coulomb crystal architecture in a Penning trap. We show that the use of global operations enables us to prepare and read out these codes using only six and ten global entangling pulses, respectively. The proposed experiment also allows the acquisition of syndrome information during readout. We provide a noise analysis for the presented protocols, estimating that we can achieve a six-fold improvement in coherence time with noise as high as $\sim 1\%$ on each entangling operation.

Implementation of standard quantum error-correction codes for solid-state qubits

In quantum error-correcting code (QECC), many quantum operations and measurements are necessary to correct errors in logical qubits. In the stabilizer formalism, which is widely used in QECC, generators $G_i (i=1,2,..)$ consist of multiples of Pauli matrices and perform encoding, decoding and measurement. In order to maintain encoding states, the stabilizer Hamiltonian $H_{\rm stab}=-\sum_i G_i$ is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian $H_{\rm stab}$ from conventional two-body Hamiltonians with Ising interaction and XY interaction by applying a pulse control method such as an NMR technique. We show that generation times of $H_{\rm stab}$ for nine-qubit code, five-qubit code and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state $|0....0>$. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.

Quantifying the Performance of Quantum Codes

We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code ([Formula: see text] and [Formula: see text], respectively) in terms of the entanglement fidelity. We also consider the concatenation of such codes ([Formula: see text]) and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes [Formula: see text], [Formula: see text] and [Formula: see text] by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five-qubit stabilizer code [Formula: see text]. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five-qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.

Quantum stabilizer codes for correlated and asymmetric depolarizing errors

We study the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors. Specifically, we consider the depolarizing noisy quantum memory channel and perform quantum error correction via the five and seven-qubit stabilizer codes. We characterize these codes by means of the entanglement fidelity as function of the error probability and the degree of memory. We show that their performances are lowered by the presence of correlations and we compute the error probability threshold values for codes effectiveness. Furthermore, we uncover that the asymmetry in the error probabilities does not affect the performance of the five-qubit code while it does affect the performance of the seven-qubit code which results less effective when considering correlated and symmetric depolarizing errors but more effective for correlated and asymmetric errors.

Efficient feedback controllers for continuous-time quantum error correction

We present an efficient approach to continuous-time quantum error correction that extends the low-dimensional quantum filtering methodology developed by van Handel and Mabuchi [e-print arXiv:quant-ph/0511221] to include error recovery operations in the form of real-time quantum feedback. We expect this paradigm to be useful for systems in which error recovery operations cannot be applied instantaneously. While we could not find an exact low-dimensional filter that combined both continuous syndrome measurement and a feedback Hamiltonian appropriate for error recovery, we developed an approximate reduced-dimensional model to do so. Simulations of the five-qubit code subjected to the symmetric depolarizing channel suggests that error correction based on our approximate filter performs essentially identically to correction based on an exact quantum dynamical model.

Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction

Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three-qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC codes are calculated. In addition, the effects of imprecision in projective measurements, collective bath, fault-tolerant repetition protocols, and level of parallelism in circuit constructions on the threshold values are also studied with emphasis on determining the optimal design for the fault-tolerant QEC circuit. These results provide insights into the fault-tolerant QEC process as well as useful information for designing the optimal fault-tolerant QEC circuit for particular physical implementation of quantum computer.

Channel kets, entangled states, and the location of quantum information

The well-known duality relating entangled states and noisy quantum channels is expressed in terms of a channel ket, a pure state on a suitable tripartite system, which functions as a pre-probability allowing the calculation of statistical correlations between, for example, the entrance and exit of a channel, once a framework has been chosen so as to allow a consistent set of probabilities. In each framework the standard notions of ordinary (classical) information theory apply, and it makes sense to ask whether information of a particular sort about one system is or is not present in another system. Quantum effects arise when a single pre-probability is used to compute statistical correlations in different incompatible frameworks, and various constraints on the presence and absence of different kinds of information are expressed in a set of all-or-nothing theorems which generalize or give a precise meaning to the concept of ``no-cloning.'' These theorems are used to discuss: the location of information in quantum channels modeled using a mixed-state environment; the $CQ$ (classical-quantum) channels introduced by Holevo; and the location of information in the physical carriers of a quantum code. It is proposed that both channel and entanglement problems be classified in terms of pure states (functioning as pre-probabilities) on systems of $p\geq 2$ parts, with mixed bipartite entanglement and simple noisy channels belonging to the category $p=3$, a five-qubit code to the category $p=6$, etc.; then by the dimensions of the Hilbert spaces of the component parts, along with other criteria yet to be determined.