Universal logic with encoded spin qubits in silicon
Aaron Weinstein, Matthew D. Reed, Aaron M. Jones, Reed W. Andrews, David Barnes, Jacob Blumoff, Larken E. Euliss, Kevin Eng, Bryan H. Fong, Sieu D. Ha, Daniel R. Hulbert, Clayton A. C. Jackson, Michael P. Jura, Tyler Keating, Joseph Kerckhoff, Andrey A. Kiselev, Justine W. Matten, Golam Sabbir, Aaron C.T. Smith, Jeffrey Wright, Matthew T. Rakher, Thaddeus D. Ladd, Matthew Borselli
Abstract Quantum computation features known examples of hardware acceleration for certain problems, but is challenging to realize because of its susceptibility to small errors from noise or imperfect control. The principles of fault tolerance may enable computational acceleration with imperfect hardware, but they place strict requirements on the character and correlation of errors 1 . For many qubit technologies 2–21 , some challenges to achieving fault tolerance can be traced to correlated errors arising from the need to control qubits by injecting microwave energy matching qubit resonances. Here we demonstrate an alternative approach to quantum computation that uses energy-degenerate encoded qubit states controlled by nearest-neighbour contact interactions that partially swap the spin states of electrons with those of their neighbours. Calibrated sequences of such partial swaps, implemented using only voltage pulses, allow universal quantum control while bypassing microwave-associated correlated error sources 1,22–28 . We use an array of six 28 Si/SiGe quantum dots, built using a platform that is capable of extending in two dimensions following processes used in conventional microelectronics 29 . We quantify the operational fidelity of universal control of two encoded qubits using interleaved randomized benchmarking 30 , finding a fidelity of 96.3% ± 0.7% for encoded controlled NOT operations and 99.3% ± 0.5% for encoded SWAP. The quantum coherence offered by enriched silicon 5–9,16,18,20,22,27,29,31–37 , the all-electrical and low-crosstalk-control of partial swap operations 1,22–28 and the configurable insensitivity of our encoding to certain error sources 28,33,34,38 all combine to offer a strong pathway towards scalable fault tolerance and computational advantage.
Abstract Quantum computation features known examples of hardware acceleration for certain problems, but is challenging to realize because of its susceptibility to small errors from noise or imperfect control. The principles of fault tolerance may enable computational acceleration with imperfect hardware, but they place strict requirements on the character and correlation of errors 1 . For many qubit technologies 2–21 , some challenges to achieving fault tolerance can be traced to correlated errors arising from the need to control qubits by injecting microwave energy matching qubit resonances. Here we demonstrate an alternative approach to quantum computation that uses energy-degenerate encoded qubit states controlled by nearest-neighbour contact interactions that partially swap the spin states of electrons with those of their neighbours. Calibrated sequences of such partial swaps, implemented using only voltage pulses, allow universal quantum control while bypassing microwave-associated correlated error sources 1,22–28 . We use an array of six 28 Si/SiGe quantum dots, built using a platform that is capable of extending in two dimensions following processes used in conventional microelectronics 29 . We quantify the operational fidelity of universal control of two encoded qubits using interleaved randomized benchmarking 30 , finding a fidelity of 96.3% ± 0.7% for encoded controlled NOT operations and 99.3% ± 0.5% for encoded SWAP. The quantum coherence offered by enriched silicon 5–9,16,18,20,22,27,29,31–37 , the all-electrical and low-crosstalk-control of partial swap operations 1,22–28 and the configurable insensitivity of our encoding to certain error sources 28,33,34,38 all combine to offer a strong pathway towards scalable fault tolerance and computational advantage.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.
The growing interest in quantum computers has triggered various approaches for the realization of a large-scale quantum processor that can eventually outperform classical high-performance computers. Quantum computers are attracting increasing attention for their potential to enormously accelerate the simulation of quantum systems and the solution of optimization problems, promising an immense impact, particularly in fields such as chemistry, medicine, and cybersecurity. Although it is still a long way to a practical quantum device, the progress over the last decades has shown encouraging results for a so far limited number of quantum bits, or qubits for short, which represent the computational units of the processor. One promising platform for the implementation of qubits is the spin of charge carriers in semiconductor quantum dots. Especially the use of purified silicon as a host material has significantly improved the performance of quantum operations. Advanced fabrication techniques from the established semiconductor industry in principle allow for the manufacturing of a large number of qubits on a small chip, making them specifically competitive from the perspective of scalability. In this thesis, we investigate challenges for electron spin qubits, particularly in the context of scaling to large qubit arrays for quantum computation. We evaluate the performance of quantum gates, present error mitigation strategies, and propose new quantum operations for two particular spin qubit implementations. In a dense array of Loss-DiVincenzo spin qubits on a chip, quantum information can be electrically controlled by microwave pulses. For neighboring qubits, capacitive couplings of nearby electrodes lead to crosstalk that affects the quantum computation. The first examination in this thesis addresses the effect of crosstalk on idling and operating spin qubits due to an interfering microwave drive. After the analysis and characterization of the performance, we suggest mitigation protocols for different constellations of important quantum gates, which can be easily implemented experimentally. We further show how to reduce crosstalk in a two-qubit gate using an asymmetric driving scheme, resulting in an enormous improvement of the gate performance, and may be used to generally overcome the obstacle of crosstalk. An additional source of errors in spin qubit registers originates from the finite residual exchange interaction between neighboring qubits. It arises due to a finite overlap of the electron wave functions and is caused by the limited speed at which the applied voltages can be controlled. We evaluate the fidelity of doing quantum operations under these circumstances with a varying number of neighboring qubits and present robust mitigation schemes with a decent performance also in the presence of noise. Besides crosstalk, another crucial observation in experiments using microwaves is the heating caused by the delivery of high-power radio frequency pulses to the quantum chip and results in a change of the qubit properties. We therefore investigate the impact of phonons on the energy levels in spin qubits to identify a temperature-dependent correction to the qubit state splitting. As microwave pulses are associated with potential errors, exchange-only qubits represent a promising alternative to achieve universal quantum computation only using baseband control. Therefore, we shift our focus towards exchange-only qubits and outline the main error sources in these systems, including residual exchange interactions between spins, charge noise from the applied voltages, and magnetic fluctuations in the material. A consequential complication of this implementation is that the quantum information can leak out of the qubit states. This effect, however, can be reduced in the presence of a finite magnetic field, as we will demonstrate in this thesis. The gate construction for exchange-only qubits usually requires more pulse steps than in the previous qubit type. Nevertheless, if exchange interactions are switched on simultaneously, this can result in faster quantum gates, as we demonstrate in this thesis. Especially the discovery of a quantum operation that introduces relative phases between computational and leakage states can be of interest for the construction of two-qubit gates and will be discussed in an example for a special operation space.
Abstract Universal quantum computation will require qubit technology based on a scalable platform1, together with quantum error correction protocols that place strict limits on the maximum infidelities for one- and two-qubit gate operations2, 3. Although various qubit systems have shown high fidelities at the one-qubit level4–10, the only solid-state qubits manufactured using standard lithographic techniques that have demonstrated two-qubit fidelities near the fault-tolerance threshold6 have been in superconductor systems. Silicon-based quantum dot qubits are also amenable to large-scale fabrication and can achieve high single-qubit gate fidelities (exceeding 99.9 per cent) using isotopically enriched silicon11, 12. Two-qubit gates have now been demonstrated in a number of systems13–15, but as yet an accurate assessment of their fidelities using Clifford-based randomized benchmarking, which uses sequences of randomly chosen gates to measure the error, has not been achieved. Here, for qubits encoded on the electron spin states of gate-defined quantum dots, we demonstrate Bell state tomography with fidelities ranging from 80 to 89 per cent, and two-qubit randomized benchmarking with an average Clifford gate fidelity of 94.7 per cent and an average controlled-rotation fidelity of 98 per cent. These fidelities are found to be limited by the relatively long gate times used here compared with the decoherence times of the qubits. Silicon qubit designs employing fast gate operations with high Rabi frequencies16, 17, together with advanced pulsing techniques18, should therefore enable much higher fidelities in the near future.
Quantum error correction (QEC) is indispensable for scalable quantum computing, but implementing it with minimal hardware overhead remains a central challenge. Large spin systems with collective degrees of freedom offer a promising route to reducing the control complexity of qubit architectures while retaining a large Hilbert space for fault-tolerant encoding. However, existing proposals for logical gates and QEC in spin ensembles generally rely on inefficient higher-order interactions. Here we introduce spin-N-Cat codes, which encode logical qubits in superpositions of spin-coherent states and generalize bosonic Cat codes to the modular subspaces of permutationally symmetric spin ensembles. The code corrects collective and individual dephasing, excitation, and decay errors. We also present an efficient physical realization in central-spin systems, such as a quantum dot, where encoding, decoding, and a universal, fault-tolerant, and bias-preserving gate set are implemented using only first-order interactions. Numerical simulations demonstrate high logical fidelity under dephasing and excitation-decay noise, independent of noise bias, and that full QEC cycles are feasible with realistic microscopic parameters. For the large collective spins available in quantum dots, this translates into a substantial extension of coherence time. Our results establish spin-N-Cat codes as a scalable, hardware-efficient approach to QEC in spin-based quantum architectures.
Recent advances in quantum error correction codes for fault-tolerant quantum computing and physical realizations of high-fidelity qubits in multiple platforms give promise for the construction of a quantum computer based on millions of interacting qubits. However, the classical-quantum interface remains a nascent field of exploration. Here, we propose an architecture for a silicon-based quantum computer processor based on complementary metal-oxide-semiconductor (CMOS) technology. We show how a transistor-based control circuit together with charge-storage electrodes can be used to operate a dense and scalable two-dimensional qubit system. The qubits are defined by the spin state of a single electron confined in quantum dots, coupled via exchange interactions, controlled using a microwave cavity, and measured via gate-based dispersive readout. We implement a spin qubit surface code, showing the prospects for universal quantum computation. We discuss the challenges and focus areas that need to be addressed, providing a path for large-scale quantum computing.
The fault-tolerant operation of logical qubits is an important requirement for realizing a universal quantum computer. Spin qubits based on quantum dots have great potential to be scaled to large numbers because of their compatibility with standard semiconductor manufacturing. Here, we show that a quantum error correction code can be implemented using a four-qubit array in germanium. We demonstrate a resonant SWAP gate and by combining controlled-Z and controlled-S−1 gates we construct a Toffoli-like three-qubit gate. We execute a two-qubit phase flip code and find that we can preserve the state of the data qubit by applying a refocusing pulse to the ancilla qubit. In addition, we implement a phase flip code on three qubits, making use of a Toffoli-like gate for the final correction step. Both the quality and quantity of the qubits will require significant improvement to achieve fault-tolerance. However, the capability to implement quantum error correction codes enables co-design development of quantum hardware and software, where codes tailored to the properties of spin qubits and advances in fabrication and operation can now come together to scale semiconductor quantum technology toward universal quantum computers.