Tensor networks are the main building blocks in a wide variety of computational sciences, ranging from many-body theory and quantum computing to probability and machine learning. Here we propose a parallel algorithm for the contraction of tensor networks using probabilistic graphical models. Our approach is based on the heuristic solution of the $μ$-treewidth deletion problem in graph theory. We apply the resulting algorithm to the simulation of random quantum circuits and discuss the extensions for general tensor network contractions.
We introduce a new open-source software library Jet, which uses task-based parallelism to obtain speed-ups in classical tensor-network simulations of quantum circuits. These speed-ups result from i) the increased parallelism introduced by mapping the tensor-network simulation to a task-based framework, ii) a novel method of reusing shared work between tensor-network contraction tasks, and iii) the concurrent contraction of tensor networks on all available hardware. We demonstrate the advantages of our method by benchmarking our code on several Sycamore-53 and Gaussian boson sampling (GBS) supr…
Matrix Product Evolution: A Method for Simulating Quantum Circuits Using Tensor Networks
BIPROGY Inc.（BIPROGY公司）；University of Tokyo（东京大学）；Niigata University（新潟大学）；Institute for Physics of Intelligence, University of Tokyo（东京大学智能物理研究所）；Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）
One of the key problems in tensor network based quantum circuit simulation is the construction of a contraction tree which minimizes the cost of the simulation, where the cost can be expressed in the number of operations as a proxy for the simulation running time. This same problem arises in a variety of application areas, such as combinatorial scientific computing, marginalization in probabilistic graphical models, and solving constraint satisfaction problems. In this paper, we reduce the computationally hard portion of this problem to one of graph linear ordering, and demonstrate how existin…
The computational advantage of noisy quantum computers has been demonstrated by sampling the bitstrings of quantum random circuits. An important issue is how the performance of quantum devices could be quantified in the so-called "supremacy regime". The standard approach is through the linear cross entropy benchmark (XEB), where the theoretical value of the probability is required for each bitstring. However, the computational cost of XEB grows exponentially. So far, random circuits of the 53-qubit Sycamore chip were verified up to 10 cycles of gates only; the XEB fidelities of deeper circuits…
With the current rate of progress in quantum computing technologies, systems with more than 50 qubits will soon become reality. Computing ideal quantum state amplitudes for circuits of such and larger sizes is a fundamental step to assess both the correctness, performance, and scaling behavior of quantum algorithms and the fidelities of quantum devices. However, resource requirements for such calculations on classical computers grow exponentially. We show that deferring tensor contractions can extend the boundaries of what can be computed on classical systems. To demonstrate this technique, we…
Classical simulation of quantum computation is necessary for studying the numerical behavior of quantum algorithms, as there does not yet exist a large viable quantum computer on which to perform numerical tests. Tensor network (TN) contraction is an algorithmic method that can efficiently simulate some quantum circuits, often greatly reducing the computational cost over methods that simulate the full Hilbert space. In this study we implement a tensor network contraction program for simulating quantum circuits using multi-core compute nodes. We show simulation results for the Max-Cut problem o…
A fermionic operator circuit is a product of fermionic operators of usually different and partially overlapping support. Further elements of fermionic operator circuits (FOCs) are partial traces and partial projections. The presented framework allows for the introduction of fermionic versions of known qudit operator circuits (QUOC), important for the simulation of strongly correlated d-dimensional systems: The multiscale entanglement renormalization ansatz (MERA), tree tensor networks (TTN), projected entangled pair states (PEPS), or their infinite-size versions (iPEPS etc.). After the definit…
We show a similarity between two different classical simulation methods for measurement based quantum computation -- one relying on a low entanglement (tree tensor network) representation of the computer's state, and the other a tensor contraction method based on the topology of the graph state. We use this similarity to show that any quantum circuit that can be efficiently simulated via tensor contraction cannot produce much entanglement.
We study the problem of finding contraction orderings on tensor networks for physical simulations using a syncretic abstract data type, the $\textit{contraction-tree}$, and explain its connection to temporal and spatial measures of tensor contraction computational complexity (nodes express time; arcs express space). We have implemented the Ratcatcher of Seymour and Thomas for determining the carving-width of planar networks, in order to offer experimental evidence that this measure of spatial complexity makes a generally effective heuristic for limiting their total contraction time.
We introduce a novel transpiler for the placement and routing of quantum circuits on arbitrary target hardware architectures. We use finite-horizon, and optionally discounted, reward functions to heuristically find a suitable placement and routing policy. We employ a finite lookahead to refine the reward functions when breaking a tie between multiple policies. We benchmark our transpiler against multiple alternative solutions and on various test sets of quantum algorithms to demonstrate the benefits of our approach.
Quantum Computing (QC) stands to revolutionize computing, but is currently still limited. To develop and test quantum algorithms today, quantum circuits are often simulated on classical computers. Simulating a complex quantum circuit requires computing the contraction of a large network of tensors. The order (path) of contraction can have a drastic effect on the computing cost, but finding an efficient order is a challenging combinatorial optimization problem. We propose a Reinforcement Learning (RL) approach combined with Graph Neural Networks (GNN) to address the contraction ordering problem…
Noise-induced contraction of MPO truncation errors in noisy random circuits and Lindbladian dynamics
Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算机科学联合中心，美国国家标准与技术研究院/马里兰大学）；Joint Quantum Institute, NIST/University of Maryland（联合量子研究所，美国国家标准与技术研究院/马里兰大学）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Department of Physics, University of Basel（巴塞尔大学物理系）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）
Tohoku University（东北大学）；Portland State University（波特兰州立大学）
It has been experimentally proven that realizing universal quantum gates using higher-radices logic is practically and technologically possible. We developed a Parallel Genetic Algorithm that synthesizes Boolean reversible circuits realized with a variety of quantum gates on qudits with various radices. In order to allow synthesizing circuits of medium sizes in the higher radix quantum space we performed the experiments using a GPU accelerated Genetic Algorithm. Using the accelerated GA we compare heuristic improvements to the mutation process based on cost minimization, on the adaptive cost o…
Quantum algorithm design usually assumes access to a perfect quantum computer with ideal properties like full connectivity, noise-freedom and arbitrarily long coherence time. In Noisy Intermediate-Scale Quantum (NISQ) devices, however, the number of qubits is highly limited and quantum operation error and qubit coherence are not negligible. Besides, the connectivity of physical qubits in a quantum processing unit (QPU) is also strictly constrained. Thereby, additional operations like SWAP gates have to be inserted to satisfy this constraint while preserving the functionality of the original ci…
Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to tensor networks with irregular geometries. Finding the best possible contraction path for such networks is a central problem, with an exponential effect on computation time and memory footprint. In this work, we implement new randomized protocols that find very high quality contraction paths for arbitrary and large tensor networks. We test our methods on a vari…
Currently, quantum computing is developing at a high speed because its high parallelism and high computing power bring new solutions to many fields. However, due to chip process technology, it is difficult to achieve full coupling of all qubits on a quantum chip, so when compiling a quantum circuit onto a physical chip, it is necessary to ensure that the two-qubit gate acts on a pair of coupled qubits by inserting swap gates. It will cause great additional cost when a large number of swap gates are inserted, leading to the execution time of quantum circuits longer. In this paper, we designed a…
Qubit routing is a key problematic related to quantum circuit compilation. It consists in rewriting a quantum circuit by adding the least possible number of instructions to make the circuit compliant with some architecture's connectivity constraints. Usually, this problem is tackled via either SWAP insertion techniques or re-synthesis of portions of the circuit using architecture aware synthesis algorithms. In this work, we propose a meta-heuristic that couples the iterative approach of SWAP insertion techniques with greedy architecture aware synthesis routines. We propose two new compilation …
We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations and thus reformulate the quantum walk in a way that circumvents the implementation of that specific oracle and which closely follows the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our num…
We present a computer-aided design flow for quantum circuits, complete with automatic layout and control logic extraction. To motivate automated layout for quantum circuits, we investigate grid-based layouts and show a performance variance of four times as we vary grid structure and initial qubit placement. We then propose two polynomial-time design heuristics: a greedy algorithm suitable for small, congestion-free quantum circuits and a dataflow-based analysis approach to placement and routing with implicit initial placement of qubits. Finally, we show that our dataflow-based heuristic genera…
Quantum circuits are time dependent diagrams describing the process of quantum computation. Usually, a quantum algorithm must be mapped into a quantum circuit. Optimal synthesis of quantum circuits is intractable and heuristic methods must be employed. With the use of heuristics, the optimality of circuits is no longer guaranteed. In this paper, we consider a local optimization technique based on templates to simplify and reduce the depth of non-optimal quantum circuits. We present and analyze templates in the general case, and provide particular details for the circuits composed of NOT, CNOT …
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 …
We study in detail the algebraic structures underlying quantum circuits generated by CNOT gates. Our results allow us to propose polynomial-time heuristics to reduce the number of gates used in a given CNOT circuit and we also give algorithms to optimize this type of circuits in some particular cases. Finally we show how to create some usefull entangled states when a CNOT circuit acts on a fully factorized state.
Quantum process tomography (QPT), where a quantum channel is reconstructed through the analysis of repeated quantum measurements, is an important tool for validating the operation of a quantum processor. We detail the combined use of an existing QPT approach based on tensor networks (TNs) and unsupervised learning with TN contraction path finding algorithms in order to use TNs of arbitrary topologies for reconstruction. Experiments were conducted on the 7-qubit IBM Quantum Falcon Processor ibmq_casablanca, where we demonstrate this technique by matching the topology of the tensor networks used…
We present a method for classically simulating quantum circuits based on the tensor contraction model of Markov and Shi (quant-ph/0511069). Using this method we are able to classically simulate the approximate quantum Fourier transform in polynomial time. Moreover, our approach allows us to formulate a condition for the composability of simulable quantum circuits. We use this condition to show that any circuit composed of a constant number of approximate quantum Fourier transform circuits and log-depth circuits with limited interaction range can also be efficiently simulated.
We study the problem of generating independent samples from the output distribution of Google's Sycamore quantum circuits with a target fidelity, which is believed to be beyond the reach of classical supercomputers and has been used to demonstrate quantum supremacy. We propose a new method to classically solve this problem by contracting the corresponding tensor network just once, and is massively more efficient than existing methods in obtaining a large number of uncorrelated samples with a target fidelity. For the Sycamore quantum supremacy circuit with $53$ qubits and $20$ cycles, we have g…
Due to several physical limitations in the realisation of quantum hardware, today's quantum computers are qualified as Noisy Intermediate-Scale Quantum (NISQ) hardware. NISQ hardware is characterized by a small number of qubits (50 to a few hundred) and noisy operations. Moreover, current realisations of superconducting quantum chips do not have the ideal all-to-all connectivity between qubits but rather at most a nearest-neighbour connectivity. All these hardware restrictions add supplementary low-level requirements. They need to be addressed before submitting the quantum circuit to an actual…
Parameterized quantum circuits (PQCs), as one of the most promising schemes to realize quantum machine learning algorithms on near-term quantum computers, have been designed to solve machine earning tasks with quantum advantages. In this paper, we explain why PQCs with different structures can achieve generative tasks in the discrete case from the perspective of probability estimation. Although different kinds of PQCs are proposed for generative tasks, the current methods often encounter the following three hurdles: (1) the mode contraction problem; (2) unexpected data are often generated with…
