Session P44: Quantum Error Correction

A universal fault-tolerant gate set for the 5-qubit quantum code

While the smallest single-error correcting classical code encodes one bit in just three, the smallest such quantum code requires five qubits to protect one qubit. Yet, the 5-qubit quantum code is widely regarded as useless when it comes to encoded quantum computation, as it supports just one non-Pauli transversal gate, the $K=SH$ gate where $H$ is Hadamard and $S$ is the phase gate. However, transversal gates, though convenient, are not all there is to fault-tolerant computation. Here we develop non-transversal, fault-tolerant logical gates for the 5-qubit quantum code, including logical controlled-Z (CZ) and logical controlled-controlled-Z (CCZ). With $K$, we can then create fault-tolerant CNOT and Toffoli gates. Together, logical Toffoli and $K$ imply that the 5-qubit code is capable of universal, fault-tolerant quantum computation. Moreover, we achieve our results without magic states. Indeed, no ancillary qubits beyond those needed for error-correction are necessary in any of our fault-tolerant constructions. We also report fault-tolerance thresholds for our new gates, calculated by exact computer simulation. In some cases, our logical gates on the 5-qubit code have better thresholds than the analogous constructions on the next smallest quantum code, the 7-qubit code.   

Quantum fault-tolerant thresholds for universal concatenated schemes

Fault-tolerant quantum computation uses ancillary qubits in order to protect logical data qubits while allowing for the manipulation of the quantum information without severe losses in coherence. While different models for fault-tolerant quantum computation exist, determining the ancillary qubit overhead for competing schemes remains a challenging theoretical problem. In this work, we study the fault-tolerance threshold rates of different models for universal fault-tolerant quantum computation. Namely, we provide different threshold rates for the 105-qubit concatenated coding scheme for universal computation without the need for state distillation. We study two error models: adversarial noise and depolarizing noise and provide lower bounds for the threshold in each of these error regimes. Establishing the threshold rates for the concatenated coding scheme will allow for a physical quantum resource comparison between our fault-tolerant universal quantum computation model and the traditional model using magic state distillation.~   

Critical parameters of a noise model that affect fault tolerant quantum computation on a single qubit

In this work, we aim to determine the parameters of a single qubit channel that can tightly bound the logical error rate of the Steane code. We do not assume any a priori structure for the quantum channel, except that it is a CPTP map and we use a concatenated Steane code to encode a single qubit. Unlike the standard Monte Carlo technique that requires many iterations to estimate the logical error rate with sufficient accuracy, we use techniques to compute the complete effect of a physical CPTP map, at the logical level. Using this, we have studied the predictive power of several physical noise metrics on the logical error rate, and show, through numerical simulations with random quantum channels, that, on their own, none of the natural physical metrics lead to accurate predictions about the logical error rate. We then show how machine learning techniques help us to explore which features of a random quantum channel are important in predicting its logical error rate.   

Doubled Color Codes

Combining protection from noise and computational universality is one of the biggest challenges in the fault-tolerant quantum computing. Topological stabilizer codes such as the 2D surface code can tolerate a high level of noise but implementing logical gates, especially non-Clifford ones, requires a prohibitively large overhead due to the need of state distillation. In this talk I will describe a new family of 2D quantum error correcting codes that enable a transversal implementation of all logical gates required for the universal quantum computing. Transversal logical gates (TLG) are encoded operations that can be realized by applying some single-qubit rotation to each physical qubit. TLG are highly desirable since they introduce no overhead and do not spread errors. It has been known before that a quantum code can have only a finite number of TLGs which rules out computational universality. Our scheme circumvents this no-go result by combining TLGs of two different quantum codes using the gauge-fixing method pioneered by Paetznick and Reichardt. The first code, closely related to the 2D color code, enables a transversal implementation of all single-qubit Clifford gates such as the Hadamard gate and the $\pi/2$ phase shift. The second code that we call a doubled color code provides a transversal T-gate, where T is the $\pi/4$ phase shift. The Clifford+T gate set is known to be computationally universal. The two codes can be laid out on the honeycomb lattice with two qubits per site such that the code conversion requires parity measurements for six-qubit Pauli operators supported on faces of the lattice. I will also describe numerical simulations of logical Clifford+T circuits encoded by the distance-3 doubled color code. Based on a joint work with Andrew Cross.   

Efficiently simulable approximations to realistic incoherent and coherent errors and their application to threshold estimation

Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code (QECC). It is not completely clear how sensitive a code's threshold is to the error model, and whether or not a Pauli channel (PC) is a good approximation for realistic non-stabilizer errors. Within the stabilizer formalism, it has been shown that for a single qubit more accurate approximations can be obtained by expanding the PC. We now examine the feasibility of employing these error approximations at the single-qubit level to obtain better estimates of a QECC's threshold. We calculate the level-1 pseudo-threshold for the Steane [[7,1,3]] code for several error models. At the logical level, the Pauli twirled channel (PTC) provides an extremely accurate approximation for incoherent channels. However, for coherent channels, the PTC severely underestimates the magnitude of the error. By computing the effective 1-qubit process matrix for the whole circuit at low error rates, it becomes clear that this behavior is due to the stronger persistence of off-diagonal entries in the coherent channels. Therefore, if the main source of error in the quantum system is coherent, reliable stabilizer simulations should employ expanded Clifford channels.   

Decoder for 3-D color codes

Transversal circuits are important components of fault-tolerant quantum computation. Several classes of quantum error-correcting codes are known to have transversal implementations of any logical Clifford operation. However, to achieve universal quantum computation, it would be helpful to have high-performance error-correcting codes that have a transversal implementation of some logical non-Clifford operation. The 3-D color codes~\footnote{H. Bomb\'{i}n, New J. Phys. {\bf 17}, 083002 (2015).} are a class of topological codes that permit transversal implementation of the logical $\pi /8$-gate. The decoding problem of a 3-D color code can be understood as a graph-matching problem on a three-dimensional lattice. Whether this class of codes will be useful in terms of performance is still an open question. We investigate the decoding problem of 3-D color codes and analyze the performance of some possible decoders.   

High-threshold decoding algorithms for the gauge color code

Gauge color codes are topological quantum error correcting codes on three dimensional lattices. They have garnered recent interest due to two important properties: (1) they admit a universal transversal gate set, and (2) their structure allows reliable error correction using syndrome data obtained from a measurement circuit of constant depth. Both of these properties make gauge color codes intriguing candidates for low overhead fault-tolerant quantum computation. Recent work by Brown et al. calculated a threshold of ~0.31\% for a particular gauge color code lattice using a simple clustering decoder and phenomenological noise. We show that we can achieve improved threshold error rates using the efficient Wootton and Loss Markov-chain Monte Carlo (MCMC) decoding. In the case of the surface code, the MCMC decoder produced a threshold close to that code's upper bound. While no upper bound is known for gauge color codes, the thresholds we present here may give a better estimate.   

Potts glass reflection of the decoding threshold for qudit quantum error correcting codes

We map the maximum likelihood decoding threshold for qudit quantum error correcting codes to the multicritical point in generalized Potts gauge glass models, extending the map constructed previously for qubit codes [1]. An $n$-qudit quantum LDPC code, where a qudit can be involved in up to $m$ stabilizer generators, corresponds to a ${\mathbb Z}_d$ Potts model with $n$ interaction terms which can couple up to $m$ spins each. We analyze general properties of the phase diagram of the constructed model, give several bounds on the location of the transitions, bounds on the energy density of extended defects (non-local analogs of domain walls), and discuss the correlation functions which can be used to distinguish different phases in the original and the dual models. \hfill\smallskip\\[0pt] [1] A A Kovalev and L P Pryadko, Quant.\ Inf.\ \& Comp.\ {\bf 15}, 0825 (2015).   

Fault-tolerant quantum computation in multiqubit block codes: performance and overhead

Fault-tolerant quantum computation requires that quantum information remain encoded in a quantum error-correcting code at all times; that a universal set of logical unitary gates and measurements is available; and that the probability of an uncorrectable error is low for the duration of the computation. Quantum computation can in principle be scaled up to unlimited size if the rate of decoherence is below a threshold. The main constructions that have been studied involve encoding each logical qubit in a separate block (either a concatenated code or a block of the surface code), which typically requires thousands of physical qubits per logical qubit, if not more. To reduce this overhead, we consider using multiqubit codes to achieve much higher storage rates. We estimate performance and overhead for certain families of codes, and ask: how large a quantum computation can be done as a function of the decoherence rate for a fixed size code block? Finally, we consider remaining open questions and limitations to this approach.   

Numerical Simulation of Coherent Error Correction

A major goal in quantum computation is the implementation of error correction to produce a logical qubit with an error rate lower than that of the underlying physical qubits. Recent experimental progress demonstrates physical qubits can achieve error rates sufficiently low for error correction, particularly for codes with relatively high thresholds such as the surface code and color code. Motivated by experimental capabilities of neutral atom systems, we use numerical simulation to investigate whether coherent error correction can be effectively used with the 7-qubit color code. The results indicate that coherent error correction does not work at the 10-qubit level in neutral atom array quantum computers. By adding more qubits there is a possibility of making the encoding circuits fault-tolerant which could improve performance.   

Noise Estimation and Adaptive Encoding for Asymmetric Quantum Error Correcting Codes

We present a technique that improves the performance of asymmetric quantum error correcting codes in the presence of biased qubit noise channels. Our study is motivated by considering what useful information can be learned from the statistics of syndrome measurements in stabilizer quantum error correcting codes (QECC). We consider the case of a qubit dephasing channel where the dephasing axis is unknown and time-varying. We are able to estimate the dephasing angle from the statistics of the standard syndrome measurements used in stabilizer QECC's. We use this estimate to rotate the computational basis of the code in such a way that the most likely type of error is covered by the highest distance of the asymmetric code. In particular, we use the $[[15, 1, 3]]$ shortened Reed-Muller code which can correct one phase-flip error but up to three bit-flip errors. In our simulations, we tune the computational basis to match the estimated dephasing axis which in turn leads to a decrease in the probability of a phase-flip error. With a sufficiently accurate estimate of the dephasing axis, our memory's effective error is dominated by the much lower probability of four bit-flips.   

Details of [[7,1,3]] Syndrome Measurements

We explore different aspects of syndrome measurements (SM) for the [[7,1,3]] quantum error correction code. This includes determining how often to apply SM, comparing the performance of different SM, rearranging the order in which SM are applied, and exploring the effects of improving SM ancilla state construction. Finally, we attempt to formulate gates and their attached SM as superoperators.