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arXiv:1708.08683 · Physical Review A

Measurement-free implementations of small-scale surface codes for quantum-dot qubits

Hamdi Ercan, Joydip Ghosh, Daniel Crow, Vickram N. Premakumar, Robert Joynt, Mark G. Friesen, S. N. Coppersmith

Published Jan 16, 2018• 14 citations
arXivHTMLPDF

Abstract

The performance of quantum-error-correction schemes depends sensitively on the physical realizations of the qubits and the implementations of various operations. For example, in quantum-dot spin qubits, readout is typically much slower than gate operations, and conventional surface-code implementations that rely heavily on syndrome measurements could therefore be challenging. However, fast and accurate reset of quantum-dot qubits, without readout, can be achieved via tunneling to a reservoir. Here we propose small-scale surface-code implementations for which syndrome measurements are replaced by a combination of Toffoli gates and qubit reset. For quantum-dot qubits, this enables much faster error correction than measurement-based schemes, but requires additional ancilla qubits and non-nearest-neighbor interactions. We have performed numerical simulations of two different coding schemes, obtaining error thresholds on the orders of ${10}^{\ensuremath{-}2}$ for a one-dimensional architecture that only corrects bit-flip errors and ${10}^{\ensuremath{-}4}$ for a two-dimensional architecture that corrects bit- and phase-flip errors.

DOI
10.1103/physreva.97.012318

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The performance of quantum-error-correction schemes depends sensitively on the physical realizations of the qubits and the implementations of various operations. For example, in quantum-dot spin qubits, readout is typically much slower than gate operations, and conventional surface-code implementations that rely heavily on syndrome measurements could therefore be challenging. However, fast and accurate reset of quantum-dot qubits, without readout, can be achieved via tunneling to a reservoir. Here we propose small-scale surface-code implementations for which syndrome measurements are replaced by a combination of Toffoli gates and qubit reset. For quantum-dot qubits, this enables much faster error correction than measurement-based schemes, but requires additional ancilla qubits and non-nearest-neighbor interactions. We have performed numerical simulations of two different coding schemes, obtaining error thresholds on the orders of 10\ensuremath−2{10}^{\ensuremath{-}2}10\ensuremath−2 for a one-dimensional architecture that only corrects bit-flip errors and 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4 for a two-dimensional architecture that corrects bit- and phase-flip errors.
Spin qubits in silicon quantum dots are one of the most promising building blocks for large scale quantum computers thanks to their high qubit density and compatibility with the existing semiconductor technologies. High fidelity single-qubit gates exceeding the threshold of error correction codes like the surface code have been demonstrated, while two-qubit gates have reached 98% fidelity and are improving rapidly. However, there are other types of error --- such as charge leakage and propagation --- that may occur in quantum dot arrays and which cannot be corrected by quantum error correction codes, making them potentially damaging even when their probability is small. We propose a surface code architecture for silicon quantum dot spin qubits that is robust against leakage errors by incorporating multi-electron mediator dots. Charge leakage in the qubit dots is transferred to the mediator dots via charge relaxation processes and then removed using charge reservoirs attached to the mediators. A stabiliser-check cycle, optimised for our hardware, then removes the correlations between the residual physical errors. Through simulations we obtain the surface code threshold for the charge leakage errors and show that in our architecture the damage due to charge leakage errors is reduced to a similar level to that of the usual depolarising gate noise. Spin leakage errors in our architecture are constrained to only ancilla qubits and can be removed during quantum error correction via reinitialisations of ancillae, which ensure the robustness of our architecture against spin leakage as well. Our use of an elongated mediator dots creates spaces throughout the quantum dot array for charge reservoirs, measuring devices and control gates, providing the scalability in the design.
A remarkable characteristic of quantum computing is the potential for reliable computation despite faulty qubits. This can be achieved through quantum error correction, which is typically implemented by repeatedly applying static syndrome checks, permitting correction of logical information. Recently, the development of time-dynamic approaches to error correction has enabled different codes and implementations that do not rely on static syndrome measurements. Here we experimentally demonstrate three time-dynamic implementations of the surface code, each offering a distinct solution to hardware design challenges faced by surface code realizations. First, we embed the surface code on a hexagonal lattice, reducing the necessary couplings per qubit from four to three. Second, we walk a surface code, swapping the role of data and measure qubits each round, achieving error correction with built-in removal of accumulated non-computational errors. Finally, we realize the surface code using iSWAP gates instead of the traditional CNOT, extending the set of viable gates for error correction without additional overhead. We measure the error suppression factor when scaling from distance-3 to distance-5 codes of Λ35,hex = 2.15(2), Λ35,walk = 1.69(6) and Λ35,iSWAP = 1.56(2), achieving state-of-the-art error suppression for each. Our work demonstrates that dynamic circuit approaches meet the demands for fault tolerance and enable alternative strategies for scalable hardware design. Typical quantum error correcting codes assign fixed roles to the underlying physical qubits. Now the performance benefits of alternative, dynamic error correction schemes have been demonstrated on a superconducting quantum processor.
The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 88%8 can be achieved.
The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.Here, we present surface code error-correction schemes using only Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.
We realize Surface Code quantum memories for nearest-neighbor qubits with always-on Ising interactions. This is done by utilizing multi-qubit gates that mimic the functionality of several gates. The previously proposed Surface Code memories rely on error syndrome detection circuits based on CNOT gates. In a two-dimensional planar architecture, to realize a two-qubit CNOT gate in the presence of couplings to other neighboring qubits, the interaction of the target qubit with its three other neighbors must cancel out. Here we present a new error syndrome detection circuit utilizing multi-qubit parity gates. In addition to speed up in the error correction cycles, in our approach, the depth of the error syndrome detection circuit does not grow by increasing the number of qubits in the logical qubit layout. We analytically design the system parameters to realize new five-qubit gates suitable for error syndrome detection in nearest-neighbor two-dimensional array of qubits. The five-qubit gates are designed such that the middle qubit is the target qubit and all four coupled neighbors are the control qubits. In our scheme, only one control parameter of the target qubits must be adjusted to realize controlled-unitary operations. The gate operations are confirmed with a fidelity of >99.9% in a simulated system consists of nine nearest-neighbor qubits.