Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a subset of one-dimensional materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. Composed of two-qubit matchgates on nearest-neighbor qubits, these constant-depth circuits are constructed based on a set of multi-matchgate identity relationships. For an $N$-spin system, the constant-depth circuit contains only $\mathcal{O}(N^2)$ CNOT gates. When compared to standard Hamiltonian simulation algorithms, our method generates circuits with order-of-magnitude fewer gates, which allows us to successfully simulate the long-time dynamics of systems with up to 5 spins on available quantum hardware. This paves the way for simulations of long-time dynamics for scientifically and technologically relevant quantum materials, enabling the observation of interesting and important atomic-level physics.
We show that low-depth random quantum circuits can be efficiently simulated by a quantum teleportation-inspired algorithm. By using logical qubits to redirect and teleport the quantum information in quantum circuits, the original circuits can be renormalized to new circuits with a smaller number of logical qubits. We demonstrate the algorithm to simulate several random quantum circuits, including 1D-chain 1000-qubit 42-depth, 2D-grid 125*8-qubit 42-depth and 2D-Bristlecone 72-qubit 32-depth circuits. Our results present a memory-efficient method with a clear physical picture to simulate low-de…
Practical Log-Depth Quantum State Preparation and Circuit Verification via Tree Tensor Network Compilation
University College London（伦敦大学学院）；Centre for Computational Science, Department of Chemistry, University College London（伦敦大学学院化学系计算科学中心）；Advanced Research Computing Centre, University College London（伦敦大学学院高级计算研究中心）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）
In breakthrough work, Bravyi, Gosset, and König (BGK) [Science, 2018] unconditionally proved that constant depth quantum circuits are more powerful than their classical counterparts. Their result is equivalent to saying that a particular family of constant depth quantum circuits takes classical circuits at least $Ω(\log n)$ depth to "simulate", in a certain sense. In our paper, we formalise their sense of simulation, which we call "possibilistic simulation" or "p-simulation", and construct explicit classical circuits that can p-simulate any depth-$d$ quantum circuit with Clifford and $t$ $T$-g…
We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware sinc…
We develop a global variable substitution method that reduces $n$-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to $3$-SAT and analyze the optimal quantum circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark $3$-SAT problems, we find that the upper bound of the circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulatio…
Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of…
We present a scalable set of universal gates and multiply controlled gates in a qudit basis through a bijective mapping from N qubits to qudits with D = 2^N levels via rotations in U(2). For each of the universal gates (H, CNOT, and T), as well as the NOT gate and multiply-controlled-Z gates, we describe a systematic approach to identifying the set of U(2) rotations required to implement each gate for any qudit of size N and with minimal use of an ancilla level. The qudit gates are analyzed in terms of the total rotation count and gate depth as the system scales with D. We apply the qudit-basi…
We focus on the depth optimization of CNOT circuits on hardwares with limited connectivity. We adapt the algorithm from Kutin et al. that implements any $n$-qubit CNOT circuit in depth at most $5n$ on a Linear Nearest Neighbour (LNN) architecture. Our proposal is a block version of Kutin et al.'s algorithm that is scalable with the number of interactions available in the hardware: the more interactions we have the less the depth. We derive better theoretical upper bounds and we provide a simple implementation of the algorithm. Overall, we achieve better depth complexity for CNOT circuits on so…
Constant-Depth Quantum Imaginary Time Evolution Using Dynamic Fan-out Circuits
Jeppesen ForeFlight（杰普森福飞）；Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology（查尔姆斯理工大学微技术和纳米科学系（MC2））；Department of Chemistry and Chemical Engineering, Chalmers University of Technology（查尔姆斯理工大学化学与化学工程系）；Center for Advanced Systems Understanding, Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心高级系统理解中心）；Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, TU Dresden（德累斯顿/莱比锡可扩展数据分析与人工智能中心，德累斯顿工业大学）
The efficient decomposition of multi-controlled gates is a significant factor in quantum compiling, both in circuit depth and T-gate count. Recent work has demonstrated that qudits have the potential to reduce resource requirements from linear to logarithmic depth and to avoid fractional phase rotations. Here we argue, based on the scaling of decoherence in high-index states, that circuit depth is not the primary factor, and that both the choice of entangling gate and interaction network topology act together to determine the spread of errors and ultimate failure rate in a circuit. We further …
Approximate Quantum Error Correction with 1D Log-Depth Circuits
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
The development of new superconducting circuits and the improvement of existing ones rely on the accurate modeling of spectral properties which are key to achieving the needed advances in qubit performance. Systematic circuit analysis at the lumped-element level, starting from a circuit network and culminating in a Hamiltonian appropriately describing the quantum properties of the circuit, is a well-established procedure, yet cumbersome to carry out manually for larger circuits. We present work utilizing symbolic computer algebra and numerical diagonalization routines versatile enough to tackl…
A significant problem for current quantum computers is noise. While there are many distinct noise channels, the depolarizing noise model often appropriately describes average noise for large circuits involving many qubits and gates. We present a method to mitigate the depolarizing noise by first estimating its rate with a noise-estimation circuit and then correcting the output of the target circuit using the estimated rate. The method is experimentally validated on the simulation of the Heisenberg model. We find that our approach in combination with readout-error correction, randomized compili…
The mapping of quantum circuits to qubits represented by ion states in a storage ring quantum computer is examined. Serial, parallel and hybrid architectures for mapping such qubits onto a storage ring quantum computer are presented. Parallelism is an important part of the SRQC architecture in terms of multiple ions, windows and lasers. A "predecessor problem" that arises in such architectures is identified and a solution is proposed. Representative numerical sizing and timing calculations are presented. The entire methodology is very general and extends to several implementation options. Open…
In this work, we propose a generalization of the current most widely used quantum computing hardware metric known as the quantum volume. The quantum volume specifies a family of random test circuits defined such that the logical circuit depth is equal to the total number of qubits used in the computation. However, such square circuit shapes do not directly relate to many specific applications for which one may wish to use a quantum computer. Based on surveying available resource estimates for known quantum algorithms, we generalize the quantum volume to a handful of representative circuit shap…
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…
We provide and experimentally demonstrate an accreditation protocol that upper-bounds the variation distance between noisy and noiseless probability distributions of the outputs of arbitrary quantum computations. We accredit the outputs of twenty-four quantum circuits executed on programmable superconducting hardware, ranging from depth nine circuits on ten qubits to depth twenty-one circuits on four qubits. Our protocol requires implementing the "target" quantum circuit along with a number of random Clifford circuits and subsequently post-processing the outputs of these Clifford circuits. Imp…
Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit
Mizuho Research & Technologies, Ltd.（瑞穗研究与技术株式会社）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）；Graduate School of Science and Technology, Keio University（庆应义塾大学理工学研究科）；Department of Applied Physics and Physico-Informatics, Keio University（庆应义塾大学应用物理与物性信息学系）
The Quantum State Preparation problem aims to prepare an $n$-qubit quantum state $|ψ_v\rangle =\sum_{k=0}^{2^n-1}v_k|k\rangle$ from the initial state $|0\rangle^{\otimes n}$, for a given unit vector $v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with $\|v\|_2 = 1$. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth and size complexity remain open when ancillary qubits are available. In this paper, we study efficient constructions of quantum circuits with $m$ ancillary qubits that can prepare…
The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using …
With the rapid progress made by industry and academia, quantum computers with dozens of qubits or even larger size are being realized. However, the fidelity of existing quantum computers often sharply decreases as the circuit depth increases. Thus, an ideal quantum circuit simulator on classical computers, especially on high-performance computers, is needed for benchmarking and validation. We design a large-scale simulator of universal random quantum circuits, often called 'quantum supremacy circuits', and implement it on Sunway TaihuLight. The simulator can be used to accomplish the following…
Generative modeling is a flavor of machine learning with applications ranging from computer vision to chemical design. It is expected to be one of the techniques most suited to take advantage of the additional resources provided by near-term quantum computers. We implement a data-driven quantum circuit training algorithm on the canonical Bars-and-Stripes data set using a quantum-classical hybrid machine. The training proceeds by running parameterized circuits on a trapped ion quantum computer, and feeding the results to a classical optimizer. We apply two separate strategies, Particle Swarm an…
We provide an $Ω(log(n))$ lower bound for the depth of any quantum circuit generating the unique groundstate of Kitaev's spherical code. No circuit-depth lower bound was known before on this code in the general case where the gates can connect qubits even if they are far away; To the best of our knowledge, this is the first time a quantum circuit-depth lower bound is given for unique ground state of a {\it gapped} local Hamiltonian. Providing a lower bound in this case seems more challenging, since such systems exhibit exponential decay of correlations and standard lower bound techniques do no…
We investigate the problem of synthesizing T-depth optimal quantum circuits over the Clifford+T gate set. First we construct a special subset of T-depth 1 unitaries, such that it is possible to express the T-depth-optimal decomposition of any unitary as product of unitaries from this subset and a Clifford (up to global phase). The cardinality of this subset is at most $n\cdot 2^{5.6n}$. We use nested meet-in-the-middle (MITM) technique to develop algorithms for synthesizing provably \emph{depth-optimal} and \emph{T-depth-optimal} circuits for exactly implementable unitaries. Specifically, for …
The quantum circuit layout (QCL) problem is to map a quantum circuit such that the constraints of the device are satisfied. We introduce a quantum circuit mapping heuristic, QXX, and its machine learning version, QXX-MLP. The latter infers automatically the optimal QXX parameter values such that the layed out circuit has a reduced depth. In order to speed up circuit compilation, before laying the circuits out, we are using a Gaussian function to estimate the depth of the compiled circuits. This Gaussian also informs the compiler about the circuit region that influences most the resulting circu…
A defining feature in the field of quantum computing is the potential of a quantum device to outperform its classical counterpart for a specific computational task. By now, several proposals exist showing that certain sampling problems can be done efficiently quantumly, but are not possible efficiently classically, assuming strongly held conjectures in complexity theory. A feature dubbed quantum speedup. However, the effect of noise on these proposals is not well understood in general, and in certain cases it is known that simple noise can destroy the quantum speedup. Here we develop a fault-t…
We derive a deterministic protocol to implement a general single-qubit POVM on near-term circuit-based quantum computers. The protocol has a modular structure, such that an $n$-element POVM is implemented as a sequence of $(n-1)$ circuit modules. Each module performs a $2$-element POVM. Two variations of the protocol are suggested, one optimal in terms of number of ancilla qubits, the other optimal in terms of number of qubit gate operations and quantum circuit depth. We use the protocol to implement $2$- and $3$-element POVMs on two publicly available quantum computing devices. The results we…
