Simulating the performance of a distance-3 surface code in a linear ion trap

Abstract

We explore the feasibility of implementing a small surface code with 9 data qubits and 8 ancilla qubits, commonly referred to as surface-17, using a linear chain of 171 Yb+ ions. Two-qubit gates can be performed between any two ions in the chain with gate time increasing linearly with ion distance. Measurement of the ion state by fluorescence requires that the ancilla qubits be physically separated from the data qubits to avoid errors on the data due to scattered photons. We minimize the time required to measure one round of stabilizers by optimizing the mapping of the two-dimensional surface code to the linear chain of ions. We develop a physically motivated Pauli error model that allows for fast simulation and captures the key sources of noise in an ion trap quantum computer including gate imperfections and ion heating. Our simulations showed a consistent requirement of a two-qubit gate fidelity of 99.9% for the logical memory to have a better fidelity than physical two-qubit operations. Finally, we perform an analysis of the error subsets from the importance sampling method used to bound the logical error rates to gain insight into which error sources are particularly detrimental to error correction.

Fault-tolerant magic state preparation with flag qubits

“…For example, trapped ion experiments have coherence times T 2 = 2T 1 on the order of 1 − 10 seconds with gate times on the order of 10µs. This suggests idle error probabilities of 10 −5 to 10 −6 , while two qubit gate infidelities are on the order of 10 −3 to 10 −4 [31,32]. On the other hand, the assumption may not hold in systems such as superconducting qubits, whose gates currently achieve infidelities near the coherence limit [33].…”

Direct measurement of Bacon-Shor code stabilizers

“…To perform error correction using Surface-17 and Bacon-Shor-13, we designed two-step lookup table decoders for both codes. The details of the Surface-17 decoder can be found at [2]. For any two-step decoder, in the first step if the syndrome shows no errors then no correction is performed; if the syndrome shows errors, then a second syndrome is measured and correction is applied based on the second syndrome.…”

“…As Surface-17 requires a large overhead of operations for logical state preparation when compared to Bacon-Shor-13, the fault-tolerant performance of the two codes will be largely dependent on the performance of state To simulate the performance of error-correcting codes on an ion trap quantum computer we need to map the codes onto a linear ion chain and compile controlled-not gates from Mølmer-Sørensen gates [32,33]. Details of operations with ion trap quantum computer can be found in [2]. Using a simulated annealing algorithm, we searched for ion chain arrangements that minimized total time for quantum error correction or average two-qubit gate time (see Appendix).…”

“…We then examine the implications of this method for small quantum error-correcting codes by comparing distance 3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped ion quantum computer. We find that for a simple circuit of prepare, error correct and measure the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.Quantum information experiments are approaching the number of qubits and operational fidelity necessary for quantum error correction to improve performance [1][2][3]. Classical error correction on quantum devices have already shown the ability to suppress introduced errors and increase memory times [4][5][6][7][8].…”

“…1 we see that the rotated surface code and the Bacon-Shor code can be considered as different choices of gauge fixings on this compass model: for the surface code, each weight-2 stabilizer is exactly a gauge operator of the corresponding type, and each weight-4 stabilizer is equivalent to fixing the parity of the product of two gauge operators on the same face; for the Bacon-Shor code, each weight-6 stabilizer is equivalent to fixing the parity of the product of three gauge operators in the same double rows or columns of qubits.The surface code is considered a promising candidate for fault-tolerant quantum computing [23][24][25]. It is also a popular choice for implementing error correction on nearterm small quantum devices [2,26,27] due to its ability to restrict all stabilizer measurements as local operations and to perform fault-tolerant syndrome extraction with a bare ancillary qubit per check operator. The ability to use bare ancillary qubits relies on a proper choice of circuit for implementing syndrome measurement of the [[9,1,3]] surface code, as illustrated in Fig.1b.…”