Scattering Processes from Quantum Simulation Algorithms for Scalar Field Theories

One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the $\mathbb{Z}_2$ surface code. Logical qubits can be encoded in a variety of ways in the surface code, based on either boundary defects, holes, or bulk twist defects. However proposed fault-tolerant implementations of the Clifford group in these schemes are limited and often require unnecessary overhead. For example, the Clifford phase gate in certain planar and hole encodings has been proposed to be implemented using costly state injection and distillation protocols. In this paper, we show that within any encoding scheme for the logical qubits, we can fault-tolerantly implement the full Clifford group by using joint measurements involving a single appropriately encoded logical ancilla. This allows us to provide new low overhead implementations of the full Clifford group in surface and color codes. It also provides the first proposed implementations of the full Clifford group in hyperbolic codes. We further use our methods to propose state-of-the art encoding schemes for small numbers of logical qubits; for example, for code distances $d = 3,5,7$, we propose a scheme using $60, 160, 308$ (respectively) physical data qubits, which allow for the full logical Clifford group to be implemented on two logical qubits. To our knowledge, this is the optimal proposal to date, and thus may be useful for demonstration of fault-tolerant logical gates in small near-term quantum computers.

Surface code is an error-correcting method that can be applied to the implementation of a usable quantum computer. At present, a promising candidate for a usable quantum computer is based on superconductor-specifically transmon. Because errors in transmon-based quantum computers appear biasedly as Z type errors, tailored surface and XZZX codes have been developed to deal with the type errors. Even though these surface codes have been suggested for lattice structures, since transmons-based quantum computers, developed by IBM, have a heavy-hexagon structure, it is natural to ask how tailored surface code and XZZX code can be implemented on the heavy-hexagon structure. In this study, we provide a method for implementing tailored surface code and XZZX code on a heavy-hexagon structure. Even when there is no bias, we obtain $$ 0.231 \%$$ as the threshold of the tailored surface code, which is much better than $$ 0.21 \%$$ and $$ 0.209 \%$$ as the thresholds of the surface code and XZZX code, respectively. Furthermore, we can see that even though a decoder, which is not the best of the syndromes, is used, the thresholds of the tailored surface code and XZZX code increase as the bias of the Z error increases. Finally, we show that in the case of infinite bias, the threshold of the surface code is $$ 0.264\%$$ , but the thresholds of the tailored surface code and XZZX code are $$ 0.296 \% $$ and $$ 0.328 \%$$ respectively.

Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers. In order to solve the problem of influence of errors on physical qubits, we propose an approximate error correction scheme that performs dimension mapping operations on surface codes. This error correction scheme utilizes the topological properties of error correction codes to map the surface code dimension to three dimensions. Compared to previous error correction schemes, the present three-dimensional surface code exhibits good scalability due to its higher redundancy and more efficient error correction capabilities. By reducing the number of ancilla qubits required for error correction, this approach achieves savings in measurement space and reduces resource consumption costs. In order to improve the decoding efficiency and solve the problem of the correlation between the surface code stabilizer and the 3D space after dimension mapping, we employ a reinforcement learning (RL) decoder based on deep Q-learning, which enables faster identification of the optimal syndrome and achieves better thresholds through conditional optimization. Compared to the minimum weight perfect matching decoding, the threshold of the RL trained model reaches 0.78%, which is 56% higher and enables large-scale fault-tolerant quantum computation.

Stabilizing quantum states in physical qubits quantum computers has been a widely explored topic in the Noisy Intermediate-Scale Quantum era. However, much of this work has focused on simulation rather than practical implementation. In this study, an experimental advancement in Bell state stabilization is presented, which utilizes surface codes for quantum error correction across three quantum computers: ibm_fez, ibm_torino, and ibm_brisbane. Our findings indicate that error correction produces an improvement of approximately 3% in accuracy for 127-qubit systems while demonstrating a more significant enhancement of around 20% for 156-qubit systems in stabilizing the Bell state with fidelity up to 0.6 in all the experiments. This paper outlines the methodology for implementing this strategy in other applications, offering a pathway to improve results (≈20%) when experimenting with superconducting quantum computers.

One of the biggest obstacles to building a large scale quantum computer is the high qubit cost of protecting quantum information. For two-dimensional architectures, the surface code has long been the leading candidate quantum memory, but can require upwards of a thousand physical qubits per logical qubit to reach algorithmically-relevant logical error rates. In this work, we introduce a hierarchical memory formed from surface codes concatenated into high-density parity check codes. These yoked surface codes are arrayed in a rectangular grid, with parity checks (yokes) measured along each row, and optionally along each column, using lattice surgery. Our construction assumes no additional connectivity beyond a nearest-neighbor square qubit grid operating at a physical error rate of 10−3. At algorithmically-relevant logical error rates, yoked surface codes use as few as one-third the number of physical qubits per logical qubit as standard surface codes, enabling moderate-overhead fault-tolerant quantum memories in two dimensions.

Quantum computing offers unprecedented speed and power for solving complex problems, but its susceptibility to errors makes error-correction codes crucial. An important metric for evaluating the performance of error-correction codes is the code distance, which measures the number of errors a code can correct. Current qubit-plane architectures are implemented using the same code distance. However, a larger code distance reduces the logical error rate but increases resources. In fact, distance-5 code uses more resources but offers higher resilience to logical errors compared to distance-3 code. We propose a qubit-plane architecture for rotated and unrotated surface codes called Qentaurus that combines various code distances of logical qubits for robust error correction while maintaining qubit efficiency. Qentaurus is designed to be highly scalable regarding computation, layout, and code distance. We performed a case study using Qentaurus with distance-3 and -7 codes to evaluate the performance. Moreover, we additionally developed a qubit placement scheme that considers the critical path of the quantum circuit to reduce the logical error rate and gate count. Our main experimental results demonstrated that Qentaurus considerably reduces the overall logical error rate (by 15.76% on average and up to 25.25%) than that of the Distance-3 Checkerboard architecture. After additionally implementing the qubit placement scheme, the average reduction increases to 13.44% (up to 39.07%). Notably, the Qentaurus-37 maintains logical error rates comparable to those of the Distance-5 Checkerboard, achieving considerable qubit efficiency. The Distance-5 Checkerboard demands 227.7% more physical qubits than that required by the Distance-3 Checkerboard, while Qentaurus-37 utilizes just 131.2% of the physical qubits required by Distance-3 the Checkerboard, marking a substantial reduction of 96.5% in the resource gap. The overall gate count is reduced by an average of 1.74% with Qentaurus alone and by 3.33% on average with the qubit placement scheme. We believe that the proposed Qentaurus and qubit placement scheme have the potential to significantly improve the error-correction performance of large-scale quantum computers in practical applications.

Scalable solutions are essential to achieving the long-term goal of building a fault-tolerant quantum computer and energy-power consumption are fundamental limiting factors for this target. Among the available types of silicon qubits, this work focuses on Flip-Flop (FF) qubits. Energy consumption and power requirements are estimated for a square array of qubits that hosts the logical qubit. The logical qubit is implemented using the rotated Surface Code (SC) for Quantum Error Correction (QEC). By using a universal set of quantum gates, the energy usage, time and power requirements for a SC cycle are estimated based on noise level, code distance and control levels. These estimates are used to provide insights into the main scaling-up challenges for quantum computer development. This is achieved by extending a thermal model that includes energy contributions from both the cryogenic components (such as the qubit array, the cryogenic control electronics, and the cryostat) and the room temperature (RT) section (RT electronics and heat dissipation systems). The maximum numbers of physical and logical qubits are provided, as well as power consumption across the different temperature sections.

Entanglement is essential for quantum information processing, but is limited by noise. We address this by developing high-yield entanglement distillation protocols with several advancements. (1) We extend the 2-to-1 recurrence entanglement distillation protocol to higher-rate n-to-(n−1) protocols that can correct any single-qubit errors. These protocols are evaluated through numerical simulations focusing on fidelity and yield. We also outline a method to adapt any classical error-correcting code for entanglement distillation, where the code can correct both bit-flip and phase-flip errors by incorporating Hadamard gates. (2) We propose a constant-depth decoder for stabilizer codes that transforms logical states into physical ones using single-qubit measurements. This decoder is applied to entanglement distillation protocols, reducing circuit depth and enabling protocols derived from high-performance quantum error-correcting codes. We demonstrate this by evaluating the circuit complexity for entanglement distillation protocols based on surface codes and quantum convolutional codes. (3) Our stabilizer entanglement distillation techniques advance quantum computing. We propose a fault-tolerant protocol for constant-depth encoding and decoding of arbitrary states in surface codes, with potential extensions to more general quantum low-density parity-check codes. This protocol is feasible with state-of-the-art reconfigurable atom arrays and surpasses the limits of conventional logarithmic depth encoders. Overall, our study integrates stabilizer formalism, measurement-based quantum computing, and entanglement distillation, advancing both quantum communication and computing.

Efforts are being made to develop quantum processors beyond planar connectivity and quantum error correction codes suitable for 3-dimensional structures. In this study, we propose 3-dimensional architectures of logical data qubits and logical ancilla qubits based on surface codes using lattice surgery in situations where 3-dimensional nearest-neighbor connectivity between physical qubits is guaranteed. We also propose a method for performing CNOT operations on the proposed architecture. We present a physical qubit structure that stacks 2-dimensional physical qubit layers and adds several additional physical qubits for lattice surgery between adjacent layers. Our design permits the fast transversal application of CNOT operations between logical qubits in a nearest-neighbor relationship on adjacent layers, which is three times faster than the speed of standard lattice surgery CNOTs. Our design also permits the performance of CNOT operations between logical qubits that are far from each other using fast SWAP operations with transversal CNOT and lattice surgery. To demonstrate the advantages of our architecture, we present the required number of qubits and time for benchmark circuits on the proposed architecture and compare the required qubits and time on the checkerboard architecture and row-type architecture which are a 2-dimensional architectures to show the advantages of our architecture.

Single-shot quantum error correction has the potential to speed up quantum computations by removing the need for multiple rounds of syndrome extraction to be fault-tolerant. By using Quantinuum's H2 trapped-ion quantum computer, we implement the [[33,1,4]] four-dimensional (4D) surface code and perform an experimental demonstration of single-shot quantum error correction with bare ancilla qubits. We conduct memory experiments comparing the two-dimensional (2D) and 4D surface codes and find that, despite differences in qubit use and syndrome extraction circuit depth, the 4D surface code matches or outperforms the 2D surface code in both the fault-tolerant and single-shot regimes.

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

We propose an efficient scheme for transferring quantum states and generating\nentangled states between two qubits of different nature. The hybrid system\nconsists a single nitrogen vacancy (NV) center and a superconducting (SC)\nqubit, which couple to an optical cavity and a microwave resonator,\nrespectively. Meanwhile, the optical cavity and the microwave resonator are\ncoupled via the electro-optic effect. By adjusting the relative parameters, we\ncan achieve high fidelity quantum state transfer as well as highly entangled\nstates between the NV center and the SC qubit. This protocol is within the\nreach of currently available techniques, and may provide interesting\napplications in quantum communication and computation with single NV centers\nand SC qubits.\n

Developing quantum systems which are robust against noise are of prime importance to the realisation of quantum technologies. Without fault tolerance, we will never be able to preserve delicate quantum states for macroscopic time scales, a necessary requirement in the construction of scalable quantum computer. Topologically ordered models offer very beautiful mechanisms for preserving quantum states. In such models, quantum information is encoded globally over a degenerate topological Hilbert space which offers a natural robustness against local environmental noise. Remarkably, topological order is present in certain realistic condensed-matter systems. This provides a platform for accessing topological order in a laboratory. Moreover, considering condensed-matter systems enables us to combine topological features with other physical effects to enhance their behaviour. In this Thesis we study the physics of topologically ordered lattice models with defects. We seek practical applications of lattice defects for the realisation of a fault-tolerant quantum computer. The first result we present shows that anyonic data of point-like lattice defects, called twists, can be found by measuring the entanglement of the ground state of the host system. The data we learn relates to the capacity of a twist defect to perform quantum computational tasks. The second result in this Thesis concerns the dynamics of the qudit toric code model with line-like defects coupled to a thermal bath. We show we can entropically inspire fragile glassy dynamics in the system. Such dynamics qualitatively improve the coherence times of quantum information encoded in the ground state of the lattice. A novel way of achieving fault-tolerant quantum computation is by producing and manipulating twist defects of topologically ordered systems. In many ways, this paradigm of topological quantum computation is analogous to quantum computation using anyons. The first area of study in this Thesis extends the analogy between anyons and twists. Specifically, we show that the anyonic data of twists in Kitaev’s toric code model can be extracted

Quantum error correction will be a necessary component towards realizing scalable quantum computers with physical qubits. Theoretically, it is possible to perform arbitrarily long computations if the error rate is below a threshold value. The two-dimensional surface code permits relatively high fault-tolerant thresholds at the ~1% level, and only requires a latticed network of qubits with nearest-neighbor interactions. Superconducting qubits have continued to steadily improve in coherence, gate, and readout fidelities, to become a leading candidate for implementation into larger quantum networks. Here we describe characterization experiments and calibration of a system of four superconducting qubits arranged in a planar lattice, amenable to the surface code. Insights into the particular qubit design and comparison between simulated parameters and experimentally determined parameters are given. Single- and two-qubit gate tune-up procedures are described and results for simultaneously benchmarking pairs of two-qubit gates are given. All controls are eventually used for an arbitrary error detection protocol described in separate work [Corcoles et al., Nature Communications, 6, 2015]

Quantum error correction (QEC) is required in quantum computers to mitigate the impact of errorson physical qubits. The goal is to optimize the neural network for high decoding performance whilemaintaining a minimalistic hardware implementation. The errors associated with decoherence can bereduced by adopting QEC schemes that encode multiple imperfect physical qubits into a logical quantumstate, similar to classical error correction. The relevance of these studies lies in the mathematicaland software modeling and implementation of corrective codes to correct several types of quantumerrors in the development and implementation of quantum algorithms for solving classes of problems ofa classical nature. The scientific novelty of this direction is expressed in the elimination of one of theshortcomings of the quantum computing process. The development of the theory and principles for constructingmodeling systems that are resistant to external interference (dependence of data distortion onnoise, dependence of the error of a quantum computing process on the measure and purity of entanglement)for modeling quantum computing is a dynamic area, as evidenced by a large number of existingmodels reflecting certain quantum computational processes and phenomena (quantum teleportation,parallelism, entanglement of quantum states) and scientific papers. Although quantum computing is notyet ready to move from theory to practice, it is nevertheless possible to reasonably guess what form aquantum computer might take, or, more importantly for programming language design, what interface itwould be possible to interact with such a quantum computer. It is natural to apply the lessons learnedfrom the programming of classical computing to quantum computing. The analysis of works in this areashowed that a new qualitative level has now been reached, which opens up promising opportunities forthe implementation of multi-qubit quantum computing. Prospects for implementation and developmentare associated not only with technological capabilities, but also with the solution of issues of buildingeffective quantum systems for solving actual mathematical problems, cryptography problems and control(optimization) problems.