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.
Quantum computing harnesses quantum laws of nature to enable new types of algorithms, not efficiently possible on traditional computers, that may lead to breakthroughs in crucial areas like materials science and chemistry. There is rapidly growing de
mand for a quantum workforce educated in the basics of quantum computing, in particular in quantum programming. However, there are few offerings for non-specialists and little information on best practices for training computer science and engineering students. In this report we describe our experience teaching an undergraduate course on quantum computing using a practical, software-driven approach. We centered our course around teaching quantum algorithms through hands-on programming, reducing the significance of traditional written assignments and relying instead on self-paced programming exercises (Quantum Katas), a variety of programming assignments, and a final project. We observed that the programming sections of the course helped students internalize theoretical material presented during the lectures. In the survey results, students indicated that the programming exercises and the final project contributed the most to their learning process. We describe the motivation for centering the course around quantum programming, discuss major artifacts used in this course, and present our lessons learned and best practices for a future improved course offering. We hope that our experience will help guide instructors who want to adopt a practical approach to teaching quantum computing and will enable more undergraduate programs to offer quantum programming as an elective.
Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcom
es or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires $Omega(r d^2)$ parity check measurements for a distance-d code defined by r parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only r measurements. In this work, we demonstrate that fault-tolerant quantum error correction can be achieved using $O(d log(d))$ measurements for any code with distance $d geq Omega(n^alpha)$ for some constant $alpha > 0$. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than r measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code.
Quantum computing exploits quantum phenomena such as superposition and entanglement to realize a form of parallelism that is not available to traditional computing. It offers the potential of significant computational speed-ups in quantum chemistry,
materials science, cryptography, and machine learning. The dominant approach to programming quantum computers is to provide an existing high-level language with libraries that allow for the expression of quantum programs. This approach can permit computations that are meaningless in a quantum context; prohibits succinct expression of interaction between classical and quantum logic; and does not provide important constructs that are required for quantum programming. We present Q#, a quantum-focused domain-specific language explicitly designed to correctly, clearly and completely express quantum algorithms. Q# provides a type system, a tightly constrained environment to safely interleave classical and quantum computations; specialized syntax, symbolic code manipulation to automatically generate correct transformations of quantum operations, and powerful functional constructs which aid composition.
We give precise quantum resource estimates for Shors algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, im
plemented within the framework of the quantum computing software tool suite LIQ$Ui|rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2lceillog_2(n)rceil+10$ qubits using a quantum circuit of at most $448 n^3 log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shors algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shors factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not least due
to a polynomial-time decoding algorithm which admits one of the highest predicted error thresholds. We consider the dependence of this threshold on underlying assumptions including different noise models, and analyze the performance of a minimum weight perfect matching (MWPM) decoding compared to a mathematically optimal maximum likelihood (ML) decoding. Our ML algorithm tracks the success probabilities for all possible corrections over time and accounts for individual gate failure probabilities and error propagation due to the syndrome measurement circuit. We present the very first evidence for the true error threshold of an optimal circuit level decoder, allowing us to draw conclusions about what kind of improvements are possible over standard MWPM.
We determine the cost of performing Shors algorithm for integer factorization on a ternary quantum computer, using two natural models of universal fault-tolerant computing: (i) a model based on magic state distillation that assumes the availability
of the ternary Clifford gates, projective measurements, classical control as its natural instrumentation set; (ii) a model based on a metaplectic topological quantum computer (MTQC). A natural choice to implement Shors algorithm on a ternary quantum computer is to translate the entire arithmetic into a ternary form. However, it is also possible to emulate the standard binary version of the algorithm by encoding each qubit in a three-level system. We compare the two approaches and analyze the complexity of implementing Shors period finding function in the two models. We also highlight the fact that the cost of achieving universality through magic states in MTQC architecture is asymptotically lower than in generic ternary case.
