By the Gottesman-Knill Theorem, the outcome probabilities of Clifford circuits can be computed efficiently. We present an alternative proof of this result for quopit Clifford circuits (i.e., Clifford circuits on collections of $p$-level systems, where $p$ is an odd prime) using Feynman's sum-over-paths technique, which allows the amplitudes of arbitrary quantum circuits to be expressed in terms of a weighted sum over computational paths. For a general quantum circuit, the sum over paths contains an exponential number of terms, and no efficient classical algorithm is known that can compute the sum. For quopit Clifford circuits, however, we show that the sum over paths takes a special form: it can be expressed as a product of Weil sums with quadratic polynomials, which can be computed efficiently. This provides a method for computing the outcome probabilities and amplitudes of such circuits efficiently, and is an application of the circuit-polynomial correspondence which relates quantum circuits to low-degree polynomials.
Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an $n$-qubit Clifford circuit $U$ can be implemented by preparing an ancillary stabilizer state $(I\otimes U)|Φ^+\rangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing $O(1)$ ancillas t…
In this work, we introduce a new circuit optimization technique to reduce the number of T gates in Clifford+T circuits by treating T gates conjugated by Clifford gates as $\fracπ{4}$-rotations around Pauli operators. The tested benchmarks shows up to $71.43\%$ and an average $42.67\%$ reduction in T-count, both surpass the best performance reported. The worst case complexity of our algorithm is $O(nk^2)$ where $n$ is the number of qubits and $k$ is the number of T gates in the original Clifford+T circuit.
We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming PH non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way we give an extension of the Gottesman-Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabiliser and non…
Quantum computing has potential to provide exponential speedups over classical computing for many important applications. However, today's quantum computers are in their early stages, and hardware quality issues hinder the scale of program execution. Benchmarking and simulation of quantum circuits on classical computers is therefore essential to advance the understanding of how quantum computers and programs operate, enabling both algorithm discovery that leads to high-impact quantum computation and engineering improvements that deliver to more powerful quantum systems. Unfortunately, the natu…
We explore the implementation of pseudo-random single-qubit rotations and multi-qubit pseudo-random circuits constructed only from Clifford gates and the T-gate, a phase rotation of pi/4. Such a gate set would be appropriate for computations performed in a fault tolerant setting. For single-qubit rotations the distribution of parameters found for unitaries constructed from Clifford plus T quickly approaches that of random rotations and require significantly fewer gates than the construction of arbitrary single-qubit rotations. For Clifford plus T pseudo-random circuits we find an exponential c…
Bravyi and Gosset recently gave classical simulation algorithms for quantum circuits dominated by Clifford operations. These algorithms scale exponentially with the number of T-gate in the circuit, but polynomially in the number of qubits and Clifford operations. Here we extend their algorithm to qudits of odd prime dimensions. We generalize their approximate stabilizer rank method for weak simulation to qudits and obtain the scaling of the approximate stabilizer rank with the number of single-qudit magic states. We also relate the canonical form of qudit stabilizer states to Gauss sum evaluat…
Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+$T$ circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number…
Modeling and simulation is essential for predicting and verifying the behavior of fabricated quantum circuits, but existing simulation methods are either impractically costly or require an unrealistic simplification of error processes. We present a method of simulating noisy Clifford circuits that is both accurate and practical in experimentally relevant regimes. In particular, the cost is weakly exponential in the size and the degree of non-Cliffordness of the circuit. Our approach is based on the construction of exact representations of quantum channels as quasiprobability distributions over…
We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS=diag(1,1,1,i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+C…
We study optimal synthesis of Clifford circuits, and apply the results to peep-hole optimization of quantum circuits. We report optimal circuits for all Clifford operations with up to four inputs. We perform peep-hole optimization of Clifford circuits with up to 40 inputs found in the literature, and demonstrate the reduction in the number of gates by about 50%. We extend our methods to the optimal synthesis of linear reversible circuits, partially specified Clifford functions, and optimal Clifford circuits with five inputs up to input/output permutation. The results find their application in …
Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based quantum computers. The method generates training data $\{X_i^{\text{noisy}},X_i^{\text{exact}}\}$ via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where $X_i^{\text{noisy}}$ and $X_i^{\text{exact}}$ are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction…
We introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. We show that our canonical forms are T-optimal in the sense that among all the single-qutrit Clifford+T circuits implementing a given operator our canonical form uses the least number of T gates. Finally, we provide an algorithm which inputs the description of an operator (as a matrix or a circuit) and constructs the canonical form for this operator. The algorithm runs in time linear in the number of T gates. Our results provide a higher…
SymFT: Universal Fault-Tolerant Quantum Circuit Simulation via Symbolic Clifford--Pauli Frames and Stabilizer Coordinates
University of Edinburgh（爱丁堡大学）；Arclight Quantum Computing Inc.（Arclight量子计算公司）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
We consider four-dimensional qudits as qubit pairs and their qudit Pauli operators as qubit Clifford operators. This introduces a nesting, $C_1^2 \subset C_2^4 \subset C_3^2$, where $C_n^m$ is the $n$th level of the $m$-dimensional qudit Clifford hierarchy. If we can convert between logical qubits and qudits, then qudit Clifford operators are qubit non-Clifford operators. Conversion is achieved by qubit fusion and qudit fission using stabilizer circuits that consume a resource state. This resource is a fused qubit stabilizer state with a fault-tolerant state preparation using stabilizer circui…
It is important for performance studies in quantum technologies to analyze quantum circuits in the presence of noise. We introduce an error probability tensor, a tool to track generalized Pauli error statistics of qudits within quantum circuits composed of qudit Clifford gates. Our framework is compatible with qudit stabilizer quantum error-correcting codes. We show how the error probability tensor can be applied in the most general case, and we demonstrate an error analysis of bipartite qudit repeaters with quantum error correction. We provide an exact analytical solution of the error statist…
In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis $\{H, P, T\}$, where $H$, $P$ and $T$ denote the Hadamard, Phase and $π/8$ gates, respectively. This basis is known as the {\it standard set} and its universality has been shown by Boykin et al. [FOCS '99]. Our normal form has several nice properties: (i) Every circuit over this basis can easily be transformed into a normal form, and (ii) Every two normal form circuits compute same unitary matrix if and only if both circuits are identical. We also show that the number of unitary operations that…
We present an initialisation method for variational quantum algorithms applicable to intermediate scale quantum computers. The method uses simulated annealing of the efficiently simulable Clifford parameter points as a pre-optimisation to find a low energy initial condition. We numerically demonstrate the effectiveness of the technique, and how it depends on Hamiltonian structure, number of qubits and circuit depth. While a range of different problems are considered, we note that the method is particularly useful for quantum chemistry problems. This presented method could help achieve a quantu…
We describe a simple algorithm for sampling $n$-qubit Clifford operators uniformly at random. The algorithm outputs the Clifford operators in the form of quantum circuits with at most $5n + 2n^2$ elementary gates and a maximum depth of $\mathcal{O}(n\log n)$ on fully connected topologies. The circuit can be output in a streaming fashion as the algorithm proceeds, and different parts of the circuit can be generated in parallel. The algorithm has an $\mathcal{O}(n^2)$ time complexity, which matches the current state of the art. The main advantage of the proposed algorithm, however, lies in its s…
Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode $k>1$ qubits in $n$ physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long $\textit{in-situ}$ computation time. In this paper, we focus on Calderbank-Shor-Steane $[\![ n,k,d ]\!]$ (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitr…
A major challenge in developing quantum computing technologies is to accomplish high precision tasks by utilizing multiplex optimization approaches, on both the physical system and algorithm levels. Loss functions assessing the overall performance of quantum circuits can provide the foundation for many optimization techniques. In this paper, we use the quadratic error loss and the final-state fidelity loss to characterize quantum circuits. We find that the distribution of computation error is approximately Gaussian, which in turn justifies the quadratic error loss. It is shown that these loss …
Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factor $\ell/m$, where $\ell…
