Machine-learning-assisted correction of correlated qubit errors in a topological code P. Baireuther1, T. E. O’Brien1, B. Tarasinski2, and C. W. J. Beenakker1 1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 2QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands December 2017 A fault-tolerant quantum computation re- gorithm. A test on a topological code (Kitaev's toric quires an efficient means to detect and correct code [14]) revealed a performance for phase-flip er- errors that accumulate in encoded quantum in- rors that was comparable to decoders based on the formation. In the context of machine learn- minimum-weight perfect matching (MWPM or \blos- ing, neural networks are a promising new ap- som") algorithm of Edmonds [15{17]. The machine proach to quantum error correction. Here we learning paradigm promises a flexibility that the clas- show that a recurrent neural network can be sic algorithms lack, both with respect to different trained, using only experimentally accessible types of topological codes and with respect to dif- data, to detect errors in a widely used topo- ferent types of errors. logical code, the surface code, with a perfor- Several groups are exploring the capabilities of a mance above that of the established minimum- neural network decoder [18{20], but existing designs weight perfect matching (or “blossom”) de- cannot yet be efficiently deployed as a decoder in a coder. The performance gain is achieved be- surface code architecture [21{23]. Two key features cause the neural network decoder can detect which are essential for this purpose are 1: The neural correlations between bit-flip (X) and phase-flip network must have a \memory", in order to be able (Z) errors. The machine learning algorithm to process repeated cycles of stabilizer measurement adapts to the physical system, hence no noise whilst detecting correlations between cycles; and 2: model is needed. The long short-term memory The network must be able to learn from measured layers of the recurrent neural network main- data, it should not be dependent on the uncertainties tain their performance over a large number of of theoretical modeling. quantum error correction cycles, making it a In this work we design a recurrent neural network practical decoder for forthcoming experimen- decoder that has both these features, and demon- tal realizations of the surface code. strate a performance improvement over a blossom de- coder in a realistic simulation of a forthcoming error correction experiment. Our decoder achieves this im- 1 Introduction provement through its ability to detect bit-flip (X) and phase-flip (Z) errors separately as well as corre- A quantum computer needs the help of a powerful lations (Y). The blossom decoder treats a Y-error as classical computer to overcome the inherent fragility a pair of uncorrelated X and Z errors, which explains of entangled qubits. By encoding the quantum in- the improved performance of the neural network. We formation in a nonlocal way, local errors can be de- study the performance of the decoder in a simplified tected and corrected without destroying the entangle- model where the Y-error rate can be adjusted inde- ment [1,2]. Since the efficiency of the quantum error pendently of the X- and Z-error rates, and measure arXiv:1705.07855v3 [quant-ph] 17 Jan 2018 correction protocol can make the difference between the decoder efficiency in a realistic model (density ma- failure and success of a quantum computation, there trix simulation) of a state-of-the-art 17-qubit surface is a major push towards more and more efficient de- code experiment (Surface-17). coders [3]. Topological codes such as the surface code, The outline of this paper is as follows. In the next which store a logical qubit in the topology of an array section2 we summarize the results from the literature of physical qubits, are particularly attractive because we need on quantum error correction with the surface they combine a favorable performance on small cir- code. The design principles of the recurrent neural cuits with scalability to larger circuits [4{9]. network that we will use are presented in Sec.3, with In a pioneering work [10], Torlai and Melko have particular attention for the need of an internal mem- shown that the data processing power of machine ory in an efficient decoder. This is one key aspect that learning (artificial neural networks [11{13]) can be differentiates our recurrent network from the feedfor- harnessed to produce a flexible, adaptive decoding al- ward networks proposed independently [18{20] (see Accepted in Quantum 2018-01-16, click title to verify 1 √ Figure 1: Schematic of the surface code. Left: N physical data qubits are arranged on a d × d square lattice (where d = N is known as the distance of the code). For each square one makes the four-fold σx or σz correlated measurement of Eq. (2). A further set of two-fold σx and σz measurements are performed on the boundary, bringing the total number of measurements to N − 1. Right: Since direct four-fold parity measurements are impractical, the measurements are instead performed by entanglement with an ancilla qubit, followed by a measurement of the ancilla in the computational basis. Both data qubits and ancilla qubits accumulate errors during idle periods (labeled I) and during gate operations (Hadamard H and cnot), which must be accounted for by a decoder. The data qubits are also entangled with the rest of the surface code by the grayed out gates. Sec.4). A detailed description of the architecture Repeated parity check measurements ~s(t) do not and training protocol is given in Sec.5. In Sec.6 we affect the qubit within this space, nor entanglement compare the performance of the neural network de- between the logical qubit states and other systems. coder to the blossom decoder for a particular circuit However, errors in the system will cause the qubit model with varying error rates. We conclude in Sec. to drift out of the logical subspace. This continuous 7 with a demonstration of the potential of machine drift is discretized by the projective measurement, be- learning for real-world quantum error correction, by coming a series of discrete jumps between subspaces decoding data from a realistic quantum simulation of H~s(t) as time t progresses. Since ~s(t) is directly mea- the Surface-17 experiment. sured, the qubit may be corrected, i.e. brought back to the initial logical subspace H~s(0). When performing this correction, a decision must be made on whether 2 Overview of the surface code ~s(t) ~s(0) to map the logical state |0iL ∈ H~s(t) to |0iL or ~s(0) To make this paper self-contained we first describe |1iL ∈ H~s(0), as no a priori relationship exists be- the operation of the surface code and formulate the tween the labels in these two spaces. If this is done in- decoding problem. The expert reader may skip di- correctly, the net action of the time evolution and cor- rectly to the next section. rection is a logical bit-flip error. A similar choice must In a quantum error correcting (QEC) code, single be made for the {|+iL, |−iL} logical states, which if logical qubits (containing the quantum information incorrect results in a logical phase-flip error. to be protected) are spread across a larger array of N noisy physical data qubits [24, 25]. The encoding is achieved by N − 1 binary parity check measure- Information about the best choice of correction (to ments on the data qubits [26]. Before these measure- most-likely prevent logical bit-flip or phase-flip errors) ments, the state of the physical system is described by is stored within the measurement vectors ~s, which a complex vector |ψi within a 2N -dimensional Hilbert detail the path the system took in state-space from space H. Each parity check measurement Mi projects H~s(0) to H~s(t). The non-trivial task of decoding, or |ψi onto one of two 2N−1-dimensional subspaces, de- extracting this correction, is performed by a classi- pendent on the outcome si of the measurement. As cal decoder. Optimal (maximum-likelihood) decod- all parity check measurements commute, the result of ing is an NP-hard problem [27], except in the pres- a single cycle of N − 1 measurements is to project ence of specific error models [28]. However, a fault- |ψi into the intersection of all subspaces H~s decided tolerant decoder need not be optimal, and polyno- by the measurements ~s = s1, . , sN−1 (si ∈ {0, 1}). mial time decoders exist with sufficient performance This is a Hilbert space of dimension 2N /2N−1 = 2, to demonstrate error mitigation on current quantum giving the required logical qubit |ψiL. hardware [5]. This sub-optimality is quantified by the Accepted in Quantum 2018-01-16, click title to verify 2 decoder efficiency [29] 3 Neural network detection of corre- (opt) D lated errors ηd = L /L , (1) D where L is the probability of a logical error per cycle The sub-optimality of the blossom decoder comes pri- (opt) using the decoder D, and L is the probability of a marily from its inability to optimally detect Pauli-Y logical error per cycle using the optimal decoder [31]. (σy) errors [29, 31, 33]. These errors correspond to a The QEC code currently holding the record for the combination of a bit-flip (X) and a phase-flip (Z) on best performance under a scalable decoder is the sur- the same qubit, and are thus treated by a MWPM face code [3{5, 16].