Leakage Suppression in the Toric Code Martin Suchara Andrew W. Cross Jay M. Gambetta IBM T. J. Watson Research Center IBM T. J. Watson Research Center IBM T. J. Watson Research Center 1101 Kitchawan Road 1101 Kitchawan Road 1101 Kitchawan Road Yorktown Heights, New York 10598 Yorktown Heights, New York 10598 Yorktown Heights, New York 10598 Email: [email protected] Email: [email protected] Email: [email protected] Abstract—Quantum codes excel at correcting local noise but Most prior studies of quantum codes do not consider fail to correct leakage faults that excite qubits to states outside errors due to leakage, and the the impact of leakage on the the computational space. Aliferis and Terhal have shown that performance of error-correcting codes has not been thoroughly an accuracy threshold exists for leakage faults using gadgets understood, particularly in the surface code. Topological codes called leakage reduction units (LRUs). However, these gadgets are known to have a 50% threshold to idealized erasure reduce the threshold and increase experimental complexity, and errors [15], [16]. However, this idealized model is far from the costs have not been thoroughly understood. We explore a variety of techniques for leakage resilience in topological codes. realistic. Moreover, the erasure correction strategy [15], [17] Our contributions are threefold. First, we develop a leakage model measures check operators that each act over a relatively large that is physically motivated and efficient to simulate. Second, we neighborhood, and this is challenging to do with a fixed planar use Monte-Carlo simulations to survey several syndrome extrac- array of qubits [18]. In less idealized leakage models, leakage tion circuits. Third, given the capability to perform 3-outcome errors do not correspond to simple erasures since they may measurements, we present a dramatically improved syndrome remain undetected or are only imprecisely located. Leakage of processing algorithm. Our simulations show that simple circuits this form has been studied for a quantum repetition code [19] with one extra CNOT per qubit reduce the accuracy threshold and for specific types of gates applied to superconducting by less than a factor of 4 when leakage and depolarizing noise qubits [20], [21]. Despite this work, the impact of leakage on rates are comparable compared to a scenario without leakage. the accuracy threshold of topological codes remains unclear. To This becomes a factor of 2 when the decoder uses 3-outcome measurements. Finally, we make the surprising observation that address this problem, we systematically study several leakage for physical error rates less than 2 × 10−4, placing LRUs after reducing circuits and develop a new syndrome processing every gate may achieve the lowest logical error rate. We expect strategy that further enhances the benefits of these circuits. that the ideas may generalize to other topological codes. II. TORIC CODE I. INTRODUCTION Large-scale quantum computers require a fault-tolerant The toric code [1] is the prototypical example of topologi- architecture based on quantum error-correcting codes. Topo- cal stabilizer codes. It is defined on a d by d square array where logical error-correcting codes [1], [2] stand out due to their d is the code distance. The left-right and top-bottom boundaries favorable properties such as local check operators, simple of the array are associated. Vertices of the array are connected syndrome extraction circuits [3], and flexible fault-tolerant to form a graph where each of the n edges carries a physical logic based on transversal gates, code deformation [4], [5], or qubit, called a code or data qubit. Each vertex and each face lattice surgery [6]. These properties endow topological codes carries a qubit used for error-correction, called a syndrome with a high accuracy threshold, estimates of which vary from or ancilla qubit (see Fig. 1a). The stabilizer S of the code is 0:67% [7] to above 1% [8]. generated by a set of check operators fAvg and fBf g that belong to the n-qubit Pauli group and are attached to each These threshold estimates assume a depolarizing noise vertex v and face f of the graph. These operators are tensor model that approximates realistic noise. However, the model products of single qubit Pauli operators X and Z. The vertex does not include so-called leakage faults that map quantum N (or star) operators Av = "2N(v) X" are X-type checks that states out of the 2-dimensional qubit subspace and into a apply an X to the qubits on the four edges N(v) incident on higher-dimensional Hilbert space, a behavior exhibited by N vertex v. The face (or plaquette) operators Bf = "2N(f) Z" many physical realizations of qubits. Leakage errors can be are Z-type checks that apply a Z to the qubits on the four edges mitigated by constructing special quantum circuits that convert N(f) on the boundary of face f. The toric code encodes a pair leakage errors into “regular” errors and may or may not simul- of logical qubits. Representatives for each class of logical Pauli taneously raise a flag to indicate the location of the leakage X and Z operators are shown in Fig. 1a. error. Gadgets that detect leakage convert it into a located loss error that is easier to correct [9]–[11]. Leakage reduction units The check operators are measured simultaneously [3] using (LRUs) convert leakage errors into regular errors but do not circuits shown in Fig. 1b and Fig. 1c. In the first step, all give an indication that leakage has occurred [12], [13]. Leakage plaquette ancillas are prepared in the j0i state and all site mitigation with LRUs was rigorously analyzed by Aliferis and ancillas in the j+i state. In the next four steps, CNOT gates Terhal [12], who showed existence of a threshold to leakage in act between each ancilla and the data qubit above, left, right, concatenated codes. Subsequent work has shown that circuits and below the ancilla, in that order. This corresponds to the can be further simplified while remaining fault-tolerant [14]. gate order used in [7], [8]. Finally, the plaquette ancillas are Z2 X1 Z dD dD Z1 Z Z Z X2 d d plaquette R R X dL dL X X dU dU X star s s data ancilla aZ aX (a) (b) (c) Fig. 1. (a) The toric code has a natural two-dimensional layout on the surface of a torus. Qubits function as either data qubits, used to store the encoded quantum state, or ancilla qubits, used to measure check operators (stabilizers) of the quantum code. Z-type check operators associate to faces (plaquettes) and X-type check operators associate to vertices (stars). Each check operator involves four data qubits and is measured using an ancilla qubit. The torus encodes a pair of qubits whose representative logical Pauli operators X1, Z1, X2, and Z2 are shown. (b,c) These circuits measure (b) plaquette and (c) star operators using four CNOT gates together with preparations and measurements. In each circuit, data qubits dD;R;L;U interact with an ancilla qubit az=x that is prepared in the state j0i or j+i = p1 (j0i + j1i). This ancilla is then measured in the basis of eigenstates of Z or X, respectively. 2 measured in the Z eigenbasis and the star ancillas in the X probability p" on each output qutrit of an ideal gate. There is a eigenbasis. These six steps constitute an error-correction cycle. relaxation process with probability p#, analogous to amplitude Since syndrome measurements may not be reliable, O(d) damping, that acts independently on each output qutrit of an error-correction cycles are performed to improve confidence ideal gate. Another important aspect of our model concerns in the syndrome. A syndrome history consists of these O(d) two-qubit gates. If one of the interacting qubits is leaked, the syndrome measurement outcomes each with 2d2 bits. other qubit is depolarized. Noisy two-qubit gates are followed by independent excitation and relaxation maps on each output The corrective operation is determined by a decoding qutrit. algorithm that processes the syndrome history. The classic approach [3] processes plaquette and star syndromes inde- The model has parameters p (depolarizing error proba- pendently to find bit and phase error corrections. A separate bility), q (syndrome measurement error probability, we set decoding graph G = (V; E) is constructed [22] for the two q = p), p" (probability of excitation outside of the com- cases. The graph is built by translating the unit cell shown putational space), and p# (probability of relaxation back to in Fig. 2a. The edges E correspond to error events in the the computational space). For convenience, we also define error-correction circuits. The horizontal edges b and d represent the relative excitation rate r = p"=p and relaxation rate qubit errors, the vertical edge a represents measurement errors s = p#=p. Elementary quantum gates behave as follows. Idle and the diagonal edges c, e, and f represent correlated errors. qubits depolarize with probability p, where we apply one of The weight of each edge approximates the negative logarithm X, Y , or Z uniformly at random. They undergo free evolution of the probability of the corresponding error. Each nontrivial and do not leak. However, an idle leaked qubit may relax to syndrome in the syndrome history introduces a defect in the the computational space with probability p#. State preparation decoding graph. A corrective operation that is likely to succeed succeeds with probability 1−p, otherwise the orthogonal state is obtained by matching pairs of defects using the minimum is prepared. With probability p" the prepared qubit leaks.