Muyuan Li

Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here we use a trapped ion system to experimentally prepare $m$-qubit GHZ states and sample the measurement results to construct $m\times m$ logical states of the $[[m^2,1,m]]$ Shor code, up to $m=7$. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the optimal code size depends on the physical error rate and we find that $m=5$ has the best performance in our system. We further realize the direct logical encoding of the $[[9,1,3]]$ Shor code on nine qubits in a thirteen-ion chain for comparison, with $98.8(1)\%$ and $98.5(1)\%$ fidelity for state $\left\vert\pm\right\rangle_L$, respectively.

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.

We simulate four quantum error correcting codes under error models inspired by realistic noise sources in near-term ion trap quantum computers: $T_2$ dephasing, gate overrotation, and crosstalk. We use this data to find preferred codes for given error parameters along with logical error biases and a pseudothreshold which compares the physical and logical gate failure rates for a CNOT gate. Using these results we conclude that Bacon-Shor-13 is the most promising near term candidate as long as the impact of crosstalk can be mitigated through other means.

Coherent errors are a dominant noise process in many quantum computing architectures. Unlike stochastic errors, these errors can combine constructively and grow into highly detrimental overrotations. To combat this, we introduce a simple technique for suppressing systematic coherent errors in low-density parity-check (LDPC) stabilizer codes, which we call stabilizer slicing. The essential idea is to slice low-weight stabilizers into two equally-weighted Pauli operators and then apply them by rotating in opposite directions, causing their overrotations to interfere destructively on the logical subspace. With access to native gates generated by 3-body Hamiltonians, we can completely eliminate purely coherent overrotation errors, and for overrotation noise of 0.99 unitarity we achieve a 135-fold improvement in the logical error rate of Surface-17. For more conventional 2-body ion trap gates, we observe an 89-fold improvement for Bacon-Shor-13 with purely coherent errors which should be testable in near-term fault-tolerance experiments. This second scheme takes advantage of the prepared gauge degrees of freedom, and to our knowledge is the first example in which the state of the gauge directly affects the robustness of a code's memory. This work demonstrates that coherent noise is preferable to stochastic noise within certain code and gate implementations when the coherence is utilized effectively.

A Bacon-Shor code is a subsystem quantum error-correcting code on an $L \times L$ lattice where the $2(L-1)$ weight-$2L$ stabilizers are usually inferred from the measurements of $(L-1)^2$ weight-2 gauge operators. Here we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. 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.

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 171Yb+ 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 logical memory to have a better fidelity than physical two-qubit operations. Finally, we perform an analysis on the error subsets from the importance sampling method used to approximate the logical error rates in this paper to gain insight into which error sources are particularly detrimental to error correction.

We present a [[7, 1, 3]] quantum error-correcting code that is able to achieve fault-tolerant syndrome measurement using one ancillary qubit per stabilizer for an error model of independent single-qubit Pauli errors. All single-qubit Pauli errors on the ancillary qubits propagate to form exclusively correctable errors on the data qubits. The situation changes for error models with two-qubit Pauli errors. We compare the level-1 logical error rates under two noise models: the standard Pauli symmetric depolarizing error model and an anisotropic error model. The anisotropic model is motivated by control errors on two-qubit gates commonly applied to trapped ion qubits. We find that one ancillary qubit per syndrome measurement is sufficient for fault-tolerance for the anisotropic error, but is not sufficient for the standard depolarizing errors. We then show how to achieve fault tolerance for the standard depolarizing errors by adding flag qubits to check for errors on select ancilary qubits. Our results on this [[7, 1, 3]] code demonstrates how physically motivated noise models may simplify fault-tolerent protocols.