Low-distance Surface Codes under Realistic Quantum Noise
We study the performance of distance-three surface code layouts under realistic multi-parameter noise models. We first calculate their thresholds under depolarizing noise. We then compare a Pauli-twirl approximation of amplitude and phase damping to amplitude and phase damping. We find the approximate channel results in a pessimistic estimate of the logical error rate, indicating the realistic threshold may be higher than previously estimated. From Monte-Carlo simulations, we identify experimental parameters for which these layouts admit reliable computation. Due to its low resource cost and superior performance, we conclude that the 17-qubit layout should be targeted in early experimental implementations of the surface code. We find that architectures with gate times in the 5–40 ns range and times of at least 1–2 s will exhibit improved logical error rates with a 17-qubit surface code encoding.
Topological quantum error-correcting codes are a leading approach to scalable fault-tolerant quantum computation Fowler092012; Lidar2013. The most practical topological code to date is the surface code, which calls for a 2-D planar qubit layout with only nearest-neighbor interactions BravyiKitaev98; Kitaev2003; RaussendorfPRL2007; FowlerPRA2009. It has been shown to allow error rates up to a threshold of approximately 1% Fowler052012; Fowler092012; Wang2011; Stephens2013. Several quantum architectures, including superconducting devices Helmer2009; Divincenzo2009 and ion traps KMW; monroe2014; TrueNJP2011; wright2012, are suitable for realizing the surface code. Recent experiments on superconducting qubits have even demonstrated error rates in the required range Martinis14.
Until recently, the threshold for the surface code has been primarily calculated for the depolarizing channel Fowler052012; Fowler092012; Wang2011; Stephens2013. Simulation of the surface code and the depolarizing channel requires only Clifford operations and Pauli measurements on stabilizer states, allowing efficient simulation on a classical computer under the Gottesman-Knill theorem Gottesman1998; Aaronson2004.
It has been shown that realistic quantum noise such as decoherence can be sufficiently approximated by a depolarizing noise model parametrized by a method such as Pauli twirling Silva2008; Nielsen_book, enabling efficient simulation. Simulations of the surface code with noise based on Pauli-twirl approximations have been performed for several superconductor architectures Ghosh2012. Other studies have achieved efficient classical simulation of realistic noise models by using Clifford gates to approximate arbitrary gates Cory012013; Cory2013 and amplitude damping Mauricio2013.
More recently, it has been shown that the surface code threshold is significantly degraded in the presence of qubit leakage in conjunction with depolarizing noise Fowler082013. It has also been shown to achieve arbitrary reliability given modest additional qubit resources under local many-qubit errors and non-local two-qubit errors Fowler14. A recent study has determined a threshold for the surface code considering correlated errors and the coupling between qubits and the environment by formulating the problem as an Ising model Jouzdani14.
In all cases, the thresholds have been calculated for a standard surface code layout. Variations of the surface code layout have been proposed Bombin07; Horsman2012 that reduce the qubit and gate resources necessary for implementation. To the best of our knowledge, the thresholds for these modified surface code layouts have not been analyzed. In addition, studies of the threshold under realistic (non-Clifford) noise models have been limited due to the exponential cost of simulation. With device error rates rapidly approaching the surface code threshold, it is timely to investigate the performance and requirements of low-distance surface code layouts for near-term experimental implementation.
In this work, we determine the threshold for distance-three surface code layouts under depolarizing and realistic noise models. We study the layouts under an amplitude and phase damping channel and an approximation of the channel using Pauli twirling Ghosh2012. Our studies demand simulation of non-Clifford operations, which requires memory exponential in the number of qubits. We use the LIQ Liquid_Documentation software architecture for our simulations. We also outline parameter regimes that enable reliable quantum error correction for low-distance surface codes and present a decoder based on a small lookup table optimized for distance-three layouts and limited classical computation.
Our paper is organized as follows. Section II briefly reviews the surface code and three layouts for the distance-three code. We introduce our decoding method, based on a small lookup table, in Section III. Section IV describes the realistic noise models and their approximations. Our experimental methodology is introduced in Section LABEL:sec_steps. In Section LABEL:sec:results, we present our surface code simulation results. Finally, we conclude in Section LABEL:sec:conclude.
Ii Low-distance Surface codes
The surface code is a stabilizer code arranged on a 2-D lattice with nearest-neighbor interactions BravyiKitaev98. It encodes a single logical qubit in a number of physical qubits that is determined by the code distance and desired layout (described below). Through repeated measurement of its stabilizer generators, the surface code in conjunction with a classical decoding algorithm can detect errors and subsequently correct up to physical errors. The distance dictates the length of the shortest undetectable error chain and in turn is also the length of the shortest logical operator. For an excellent review of the surface code, we refer the reader to Fowler092012.
ii.1 25-qubit Layout
We study three different distance layouts, shown in Figure 1. We begin by discussing the standard layout, referred to as Surface-25, shown in Figure 1(a). It uses a square grid of qubits with a smooth and rough boundary Kitaev2003. For , the grid contains qubits of which data qubits (large white circles) are used to encode the logical qubit and syndrome qubits (small black circles) are used to extract the error syndromes by way of stabilizer measurements.
|(a) Surface-25||(b) Surface-17||(c) Surface-13|
Surface-25 is simultaneously stabilized by the group of stabilizer generators listed in Table 2. In Fig. 1, the stabilizers are represented by light (yellow) patches and the stabilizers are represented by dark (green) patches, where each patch represents a tensor product of (or ) operators on the data qubits surrounding the patch.
A logical operator is defined as a chain of physical operations between two data qubits on opposite smooth boundaries (top and bottom edges). The chain is allowed to cross any stabilizer patch and follow any edge of an stabilizer patch. A logical operator is defined analogously as a chain of physical operations between two data qubits on opposite rough boundaries (left and right edges). Table 2 lists one possible logical and operator. There are equivalent logical operators for each logical Pauli operator ( and ), where is the number of stabilizer generators for the given surface code. Since and commute with all of the stabilizers and cannot be written as a product of them, logical errors, which come in the form of logical operators, cannot be detected by the code.
The surface code detects errors through the eigenvalues of the stabilizers. A bit-flip (phase-flip) on a data qubit will change the eigenvalue of adjacent () stabilizers. To extract an eigenvalue, also referred to as an error syndrome, a given stabilizer is measured. Figure 2 shows the standard quantum circuit for measuring the stabilizers Fowler092012; Stephens2013, where data qubit corresponds to the top (north) qubit and corresponds to the bottom (south) qubit of each diamond patch in Fig. 1(a).
The circuit begins with CNOT gates that propagate error information from the data qubits ,,, to the syndrome qubit (black circle). CNOT gates are performed in the order: top (); left (); right (); bottom (). Cyclic orders, such as a clockwise or counter-clockwise, i.e., , fail to maintain commutation of nearby stabilizers, which in turn can cause random measurement outcomes Fowler092012. Thus the order of CNOT gates is required to follow an “S” or “Z” shape.
The syndrome qubit is then measured to extract the eigenvalue of the stabilizer. These error syndromes are input to a classical decoding algorithm to determine an appropriate correction operator. Details of our decoding algorithm are given in Section III. The total number of operations in a given round of stabilizer measurements for the surface code is given in Table 1.
ii.2 13- and 17-qubit Layouts
The number of qubits in Surface-25 can be reduced while maintaining the same code distance by rotating it clockwise by degrees and removing the four corner data qubits Bombin07; Horsman2012, shown in Fig. 1(b). The number of data qubits is reduced from to and the number of syndrome qubits is reduced to for a total of qubits. We call this layout Surface-17. The stabilizer generators contain weight-4 and weight-2 stabilizers (Table 2). Figure 3(a) shows the circuit for a simultaneous weight-4 and weight-2 stabilizer measurement.
A further reduction in qubits can be obtained by reusing the syndrome qubits Horsman2012. Surface-13 uses only syndrome qubits as shown in Fig. 1(c). Each syndrome qubit is used twice, once for stabilizer measurement and once for stabilizer measurement. Figure 3(b) contains the corresponding circuit for measuring a weight-4 stabilizer followed by a weight-2 stabilizer. Surface-13 reduces the number of qubits but increases the depth of a round by 4 time steps. The depth and number of operations required for one round of the surface code for Surface-17 and 13 are given in Table 1. The stabilizers and logical operations for these two layouts are listed in Table 2.
Despite having fewer stabilizers, Surface-17 and Surface-13 still remain distance-three surface codes Bombin07; Horsman2012. Due to their reduction in resources by 32–48%, these layouts are promising candidates for early experimental implementation. In Section LABEL:sec:results, we determine which layout is most promising based on its threshold and resource costs.
Iii Decoding Method
A standard method for mapping error syndromes to the most probable error chain is the minimum weight perfect matching algorithm Fowler052012; Edmonds65; Edmonds65b. It requires time for detection events if executed serially, and time if executed in parallel Fowler072013. The algorithm independently corrects and errors by identifying the most likely error chain for each type such that the total chain weight is minimal. The algorithm has recently been extended to handle correlations between and errors, in which case the chains are not constructed independently Fowler13100863. Corrections are applied along the chain(s). If after correction a chain of errors connecting two smooth (rough) boundaries remains, then a logical error has occurred. If errors are assumed to be independent, then long chains will be exponentially unlikely.
iii.1 Lookup Table Decoder
In this work we target first-generation implementations of a single qubit protected by a small surface code. While the classical time and space requirements of the minimum weight perfect matching algorithm are modest, we further reduce the classical computational overhead by designing a lookup table based on the algorithm that can be implemented on a small classical device. Our lookup table is designed to find the most probable low-weight error chain from a short history of error syndromes.
Consider the set of error syndromes that indicate an error after one full (noisy) round of the surface code, that is, those indicating a eigenvalue. Based on the error syndrome locations, the decoder determines the probable data-qubit error locations. For example, consider a error on qubit in Surface-17 (Fig. 1(b)). Given that no other errors occur, after one round of the surface code syndrome qubits and will indicate an error. The decoder will determine the shortest error chain connecting these two syndromes includes data qubit . To correct the error chain, will be applied.
As another example, consider an error on qubit . It will cause syndrome to indicate an error. Since syndromes and do not indicate errors, the decoder will infer an error on either data qubit or . In this case, the decoder can correct either or since is a stabilizer.
An error syndrome may also occur due to a measurement error. However, the decoder may interpret it as a data-qubit error. For example, consider a measurement error on qubit . The decoder will either apply or to “correct” the error, thereby adding an error to a clean data qubit.
To improve identification of actual data-qubit errors, inference is performed based on several rounds of stabilizer measurements Fowler052012. Consider performing rounds of the surface code consecutively. Instead of storing the syndromes for each round, we store the locations in time and space of the syndromes whose values change, or “flip”, between the current and previous round.
For rounds, this requires storing a 3-D space-time array containing at most values, where is the maximum number of syndrome changes in a round. We refer to this 3-D array as the syndrome volume, where dimension represents time. The goal is to determine a correction operator (a product of and/or operators) based on the syndrome volume such that the number of errors remaining after correction is minimized, in turn reducing the chance of forming a logical error chain.
Our lookup table is based on the fact that short error chains are more likely than long chains. Assuming a syndrome volume contains rounds, we construct a lookup table based on the following rules (Figure 4 shows the rules visually):
If the same syndrome flips twice in two consecutive rounds, the pair (in time) of syndromes is ignored since it most likely indicates a measurement error.
If a pair (in space) of neighboring syndromes flips in the same round, a correction on the data qubit between the pair is applied.
If a syndrome flips in round and its neighboring syndrome flips in round , a correction on the data qubit between the pair (in time) is applied.
If a syndrome flips only once and in a round other than the last, a correction is applied to a data qubit on the boundary such that the data qubit is not between two stabilizers that did not indicate a syndrome.
If a single syndrome flips only once and in the last round, the information is kept until the next round of error correction. No correction based on this syndrome is applied. In this case the location of the error, if any, is inconclusive without another round of syndrome measurements.
We decode by checking the above rules in order and determining the set of data-qubit error locations. We then switch the order of rules 2 and 3 and determine another set of possible error locations. We correct based on the set with fewer error locations, since fewer errors are more likely. Here we assume that .
These rules are equivalent to the minimum weight perfect matching algorithm applied to only neighboring-syndrome pairs, with uniform weight for the same distance. Since our surface codes are small, performance of the code does not improve when decoding considers more distant pairs.
We encode these rules into a lookup table. The lookup table maps the syndrome volume of measurement flips to a set of probable errors on the data qubits. The table requires constant time and space, where is the number of data qubits.
iii.2 Improved Stabilizer Measurement Circuits
In our simulations of Surface-13 and 17 under noise, we find that using the same CNOT ordering for both - and -type stabilizer measurements could result in a single error on a syndrome qubit, leading to a logical or error (details on noise are given in Sec. IV). Figure 5(a) shows an example. A error on a -stabilizer syndrome qubit after the first two CNOT gates propagates onto two horizontally aligned data qubits. Since our surface codes require only three data qubits to complete a logical error chain, the next round of syndrome measurements will incorrectly diagnose a error on the third qubit, leading to a logical error chain.
To prevent the creation of a logical error, we propose to measure - and -type stabilizers in different orders. The sequence for stabilizers is the same as in Figure 2. We modify the order of CNOTs in stabilizers as: top right (); bottom right (); top left (); bottom left () (Figure 5(b)). This order maintains the alignment of qubits and such that they are perpendicular to the direction of the corresponding logical chain. It also preserves the commutation relations as well as the circuit depth and size. Fig. 5 shows an example where two errors map to a single error with the new order, versus a logical error with the old order. We use this new order for all simulations in this paper.
Iv Noise models
In this section, we present the noise models considered in our surface code simulations. We review two noise models that can be simulated efficiently on a classical computer (depolarizing and Pauli-twirl approximation) and one noise model that requires exponential memory to simulate (amplitude and phase damping).
iv.1 Symmetric and Asymmetric Depolarizing Channels
The depolarizing channel () is a standard quantum noise model in which a qubit becomes depolarized with a given probability . This channel transforms a density matrix of a single qubit as
where . In this model, a qubit suffers from discrete Pauli bit-flip (), phase-flip (), or bit-and-phase flip () errors with probabilities , , and , respectively. When , this channel is called a symmetric depolarizing channel. When the probabilities are independent, the model is called an asymmetric depolarizing channel.
iv.2 Amplitude and Phase Damping Channel
The amplitude damping channel () characterizes the behavior of energy dissipation of the quantum system, including spontaneous emission of a photon from a qubit. This channel transforms the density matrix of a single qubit as
and is the probability of a qubit emitting a single photon.
Figure LABEL:fig_ad expresses amplitude damping of a single qubit in the form of a quantum circuit where an ancilla qubit is used to represent the environment and Nielsen_book. The input is an arbitrary single-qubit state and the output state is given by
where is a normalization constant. The probabilities of measuring 0 and 1 are and , respectively.