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We show how to perform a fault-tolerant universal quantum computation in 2D architectures using only transversal unitary operators and local syndrome measurements. Our approach is based on a doubled version of the 2D color code. It enables a transversal implementation of all logical gates in the Clifford+T basis using the gauge fixing method proposed recently by Paetznick and Reichardt. The gauge fixing requires six-qubit parity measurements for Pauli operators supported on faces of the honeycomb lattice with two qubits per site. Doubled color codes are promising candidates for the experimental demonstration of logical gates since they do not require state distillation. Secondly, we propose a Maximum Likelihood algorithm for the error correction and gauge fixing tasks that enables a numerical simulation of logical circuits in the Clifford+T basis. The algorithm can be used in the online regime such that a new error syndrome is revealed at each time step. We estimate the average number of logical gates that can be implemented reliably for the smallest doubled color code and a toy noise model that includes depolarizing memory errors and syndrome measurement errors.
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We present a family of quantum error-correcting codes that support a universal set of transversal logic gates using only local operations on a two-dimensional array of physical qubits. The construction is a subsystem version of color codes where gauge fixing through local measurements dynamically determines which gates are transversal. Although the operations are local, the underlying code is not topological in structure, which is how the construction circumvents no-go constraints imposed by the Bravyi-Konig and Pastawski-Yoshida theorems. We provide strong evidence that the encoding has no error threshold in the conventional sense, though it is still possible to have logical gates with error probability much lower than that of physical gates.
Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.
The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be used with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. We also prove that 1-flag stabilizer measurement circuits are sufficient to preserve the full code distance, which may be used to find simpler syndrome extraction circuits of the color code.
We have constructed a pulsed laser system for the manipulation of cold Rb atoms. The system combines optical telecommunications components and frequency doubling to generate light at 780 nm. Using a fast, fibre-coupled intensity modulator, output from a continuous laser diode is sliced into pulses with a length between 1.3 and 6.1 ns and a repetition frequency of 5 MHz. These pulses are amplified using an erbium-doped fibre amplifier, and frequency-doubled in a periodically poled lithium niobate crystal, yielding a peak power up to 12 W. Using the resulting light at 780 nm, we demonstrate Rabi oscillations on the F = 2 <-> F=3-transition of a single 87Rb atom.
Estimating and reducing the overhead of fault tolerance (FT) schemes is a crucial step toward realizing scalable quantum computers. Of particular interest are schemes based on two-dimensional (2D) topological codes such as the surface and color codes which have high thresholds but lack a natural implementation of a non-Clifford gate. In this work, we directly compare two leading FT implementations of the T gate in 2D color codes under circuit noise across a wide range of parameters in regimes of practical interest. We report that implementing the T gate via code switching to a 3D color code does not offer substantial savings over state distillation in terms of either space or space-time overhead. We find a circuit noise threshold of 0.07(1)% for the T gate via code switching, almost an order of magnitude below that achievable by state distillation in the same setting. To arrive at these results, we provide and simulate an optimized code switching procedure, and bound the effect of various conceivable improvements. Many intermediate results in our analysis may be of independent interest. For example, we optimize the 2D color code for circuit noise yielding its largest threshold to date 0.37(1)%, and adapt and optimize the restriction decoder finding a threshold of 0.80(5)% for the 3D color code with perfect measurements under Z noise. Our work provides a much-needed direct comparison of the overhead of state distillation and code switching, and sheds light on the choice of future FT schemes and hardware designs.
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