Simulating Quantum Circuits Research Articles

Hybrid Approaches for Efficient Simulations of 3-Qubit Quantum Fourier Transform (QFT) Circuit Using Quick Quantum Circuit Simulation (QQCS)

The research devised efficient methods for simulating 3-qubit Quantum Fourier Transform (QFT) circuits using Quick Quantum Circuit Simulation (QQCS). The hybrid methodologies suggested as a solution for efficiently simulating the circuit involve the combination of decision diagrams and property exploitation techniques. This paper incorporated two methods based on decision diagrams: the reordering trick and decision diagram approximations, template-based optimization, and linear reversible circuit synthesis for property exploitation. The proposed approaches significantly improved and optimized quantum algorithms and hardware by aiming to simulate quantum circuits accurately and quickly. Simulations using QQCS proved the effectiveness of these strategies, which were then compared to the original circuit. The results yielded valuable insights into enhancing simulation efficiency while upholding circuit accuracy.

LIMDD: A Decision Diagram for Simulation of Quantum Computing Including Stabilizer States

Efficient methods for the representation and simulation of quantum states and quantum operations are crucial for the optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing relevant aspects of quantum systems, but their limits are not well understood. In this work, we investigate and bridge the gap between existing DD-based structures and the stabilizer formalism, an important tool for simulating quantum circuits in the tractable regime. We first show that although DDs were suggested to succinctly represent important quantum states, they actually require exponential space for certain stabilizer states. To remedy this, we introduce a more powerful decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that the set of quantum states represented by poly-sized LIMDDs strictly contains the union of stabilizer states and other decision diagram variants. Finally, there exist circuits which LIMDDs can efficiently simulate, while their output states cannot be succinctly represented by two state-of-the-art simulation paradigms: the stabilizer decomposition techniques for Clifford + T circuits and Matrix-Product States. By uniting two successful approaches, LIMDDs thus pave the way for fundamentally more powerful solutions for simulation and analysis of quantum computing.

Efficient and Portable Einstein Summation in SQL

Computational problems ranging from artificial intelligence to physics require efficient computations of large tensor expressions. These tensor expressions can often be represented in Einstein notation. To evaluate tensor expressions in Einstein notation, that is, for the actual Einstein summation, usually external libraries are used. Surprisingly, Einstein summation operations on tensors fit well with fundamental SQL constructs. We show that by applying only four mapping rules and a simple decomposition scheme using common table expressions, large tensor expressions in Einstein notation can be translated to portable and efficient SQL code. The ability to execute large Einstein summation queries opens up new possibilities to process data within SQL. We demonstrate the power of Einstein summation queries on four use cases, namely querying triplestore data, solving Boolean satisfiability problems, performing inference in graphical models, and simulating quantum circuits. The performance of Einstein summation queries, however, depends on the query engine implemented in the database system. Therefore, supporting efficient Einstein summation computations in database systems presents new research challenges for the design and implementation of query engines.

Classical simulation of variational quantum classifiers using tensor rings

In recent times, Variational Quantum Circuits (VQC) have been widely adopted to different tasks in machine learning such as Combinatorial Optimization and Supervised Learning. With the growing interest, it is pertinent to study the boundaries of the classical simulation of VQCs to effectively benchmark the algorithms. Classically simulating VQCs can also provide the quantum algorithms with a better initialization reducing the amount of quantum resources needed to train the algorithm. Even though Matrix Product State representations have been extensively used for quantum state approximation, their capacity is limited in simulating quantum circuits due to the exponential complexity in circuit depth. This manuscript proposes an algorithm that compresses the quantum state within a circuit using a noisy tensor ring representation which allows for the implementation of VQC based algorithms on a classical simulator at a fraction of the usual storage and computational complexity. Using the tensor ring approximation of the input quantum state, we propose a method that applies the parametrized unitary operations while retaining the low-rank structure of the tensor ring corresponding to the transformed quantum state, providing an exponential improvement of storage and computational time in the number of qubits and layers. This approximation is used to implement the tensor ring VQC (TRVQC) for the task of supervised learning on Iris and MNIST datasets to demonstrate the performance of the proposed method compared with the implementations from classical simulator using Matrix Product States (MPS). TRVQC has a test accuracy of 82.63% compared to the benchmark of 83.68% on Iris dataset whereas the former outperforms the latter on a reduced MNIST dataset with TRVQC having an accuracy of 83.73% compared to the benchmark 81.02%, showcasing the comparable performance of the proposed algorithm with the MPS framework.

Density-Matrix Renormalization Group Algorithm for Simulating Quantum Circuits with a Finite Fidelity

We develop a density-matrix renormalization group (DMRG) algorithm for the simulation of quantum circuits. This algorithm can be seen as the extension of time-dependent DMRG from the usual situation of hermitian Hamiltonian matrices to quantum circuits defined by unitary matrices. For small circuit depths, the technique is exact and equivalent to other matrix product state (MPS) based techniques. For larger depths, it becomes approximate in exchange for an exponential speed up in computational time. Like an actual quantum computer, the quality of the DMRG results is characterized by a finite fidelity. However, unlike a quantum computer, the fidelity depends strongly on the quantum circuit considered. For the most difficult possible circuit for this technique, the so-called "quantum supremacy" benchmark of Google Inc. , we find that the DMRG algorithm can generate bit strings of the same quality as the seminal Google experiment on a single computing core. For a more structured circuit used for combinatorial optimization (Quantum Approximate Optimization Algorithm or QAOA), we find a drastic improvement of the DMRG results with error rates dropping by a factor of 100 compared with random quantum circuits. Our results suggest that the current bottleneck of quantum computers is their fidelities rather than the number of qubits.

Simulation Paths for Quantum Circuit Simulation With Decision Diagrams What to Learn From Tensor Networks, and What Not

Simulating quantum circuits on classical computers is a notoriously hard, yet increasingly important task for the development and testing of quantum algorithms. In order to alleviate this inherent complexity, efficient data structures and methods, such as tensor networks and decision diagrams, have been proposed. However, their efficiency heavily depends on the order in which the individual computations are performed. For tensor networks, the order is defined by so-called contraction plans and a plethora of methods has been developed to determine suitable plans. On the other hand, simulation based on decision diagrams is mostly conducted in a straight-forward, i.e., sequential, fashion thus far. In this work, we study the importance of the path that is chosen when simulating quantum circuits using decision diagrams and show, conceptually and experimentally, that choosing the right simulation path can make a vast difference in the efficiency of classical simulations using decision diagrams. We propose an open-source framework (available at github.com/cda-tum/ddsim) that not only allows to investigate of dedicated simulation paths but also to reuse of existing findings, e.g., obtained from determining contraction plans for tensor networks. Experimental evaluations show that translating strategies from the domain of tensor networks may yield speedups of several factors compared to the state of the art. Furthermore, we design a dedicated simulation path heuristic that allows to improve the performance even further—frequently yielding speedups of several orders of magnitude. Finally, we provide an extensive discussion on what can be learned from tensor networks and what cannot.

Simulating quantum circuits using tree tensor networks

We develop and analyze a method for simulating quantum circuits on classical computers by representing quantum states as rooted tree tensor networks. Our algorithm first determines a suitable, fixed tree structure adapted to the expected entanglement generated by the quantum circuit. The gates are sequentially applied to the tree by absorbing single-qubit gates into leaf nodes, and splitting two-qubit gates via singular value decomposition and threading the resulting virtual bond through the tree. We theoretically analyze the applicability of the method as well as its computational cost and memory requirements, and identify advantageous scenarios in terms of required bond dimensions as compared to a matrix product state representation. The study is complemented by numerical experiments for different quantum circuit layouts up to 37 qubits.

Experimental Simulation of Larger Quantum Circuits with Fewer Superconducting Qubits.

Although near-term quantum computing devices are still limited by the quantity and quality of qubits in the so-called NISQ era, quantum computational advantage has been experimentally demonstrated. Moreover, hybrid architectures of quantum and classical computing have become the main paradigm for exhibiting NISQ applications, where low-depth quantum circuits are repeatedly applied. In order to further scale up the problem size solvable by the NISQ devices, it is also possible to reduce the number of physical qubits by "cutting" the quantum circuit into different pieces. In this work, we experimentally demonstrated a circuit-cutting method for simulating quantum circuits involving many logical qubits, using only a few physical superconducting qubits. By exploiting the symmetry of linear-cluster states, we can estimate the effectiveness of circuit-cutting for simulating up to 33-qubit linear-cluster states, using at most 4 physical qubits for each subcircuit. Specifically, for the 12-qubit linear-cluster state, we found that the experimental fidelity bound can reach as much as 0.734, which is about 19% higher than a direct implementation on the same 12-qubit superconducting processor. Our results indicate that circuit-cutting represents a feasible approach of simulating quantum circuits using much fewer qubits, while achieving a much higher circuit fidelity.

Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions

We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.

Pulse-level noisy quantum circuits with QuTiP

The study of the impact of noise on quantum circuits is especially relevant to guide the progress of Noisy Intermediate-Scale Quantum (NISQ) computing. In this paper, we address the pulse-level simulation of noisy quantum circuits with the Quantum Toolbox in Python (QuTiP). We introduce new tools in qutip-qip, QuTiP's quantum information processing package. These tools simulate quantum circuits at the pulse level, leveraging QuTiP's quantum dynamics solvers and control optimization features. We show how quantum circuits can be compiled on simulated processors, with control pulses acting on a target Hamiltonian that describes the unitary evolution of the physical qubits. Various types of noise can be introduced based on the physical model, e.g., by simulating the Lindblad density-matrix dynamics or Monte Carlo quantum trajectories. In particular, the user can define environment-induced decoherence at the processor level and include noise simulation at the level of control pulses. We illustrate how the Deutsch-Jozsa algorithm is compiled and executed on a superconducting-qubit-based processor, on a spin-chain-based processor and using control optimization algorithms. We also show how to easily reproduce experimental results on cross-talk noise in an ion-based processor, and how a Ramsey experiment can be modeled with Lindblad dynamics. Finally, we illustrate how to integrate these features with other software frameworks.

Q<sup>2</sup>Chemistry: A quantum computation platform for quantum chemistry

Quantum computers provide new opportunities for quantum chemistry. In this article,we present a versatile, extensible, and efficient software package, named Q<sup>2</sup>Chemistry, for developing quantum algorithms and quantum inspired classical algorithms in the field of quantum chemistry. In Q<sup>2</sup>Chemistry, the wave function and Hamiltonian can be conveniently mapped into the qubit space, then quantum circuits can be generated corresponding to a specific quantum algorithm already implemented in the package or newly developed by the users. The generated circuits can be dispatched to either a physical quantum computer, if available, or to the internal virtual quantum computer realized by simulating quantum circuits on classical computers. As demonstrated by our benchmark simulations, Q<sup>2</sup>Chemistry achieves excellent performance in simulating medium scale quantum circuits using the matrix product state algorithm. Applications of Q<sup>2</sup>Chemistry to simulate molecules and periodic systems are given with performance analysis.

Probabilistic simulation of quantum circuits using a deep-learning architecture

The fundamental question of how to best simulate quantum systems using conventional computational resources lies at the forefront of condensed matter and quantum computation. It impacts both our understanding of quantum materials and our ability to emulate quantum circuits. Here we present an exact formulation of quantum dynamics via factorized generalized measurements which maps quantum states to probability distributions with the advantage that local unitary dynamics and quantum channels map to local quasistochastic matrices. This representation provides a general framework for using state-of-the-art probabilistic models in machine learning for the simulation of quantum many-body dynamics. Using this framework, we have developed a practical algorithm to simulate quantum circuits using an attention network based on a powerful neural network ansatz responsible for the most recent breakthroughs in natural language processing. We demonstrate our approach by simulating circuits that build Greenberger-Horne-Zeilinger and linear graph states of up to 60 qubits, as well as a variational quantum eigensolver circuit for preparing the ground state of the transverse field Ising model on several system sizes. Our methodology constitutes a modern machine learning approach to the simulation of quantum physics with applicability both to quantum circuits as well as other quantum many-body systems.

Simulating noisy quantum circuits with matrix product density operators

Simulating quantum circuits with classical computers requires resources growing exponentially in terms of system size. Real quantum computer with noise, however, may be simulated polynomially with various methods considering different noise models. In this work, we simulate random quantum circuits in 1D with Matrix Product Density Operators (MPDO), for different noise models such as dephasing, depolarizing, and amplitude damping. We show that the method based on Matrix Product States (MPS) fails to approximate the noisy output quantum states for any of the noise models considered, while the MPDO method approximates them well. Compared with the method of Matrix Product Operators (MPO), the MPDO method reflects a clear physical picture of noise (with inner indices taking care of the noise simulation) and quantum entanglement (with bond indices taking care of two-qubit gate simulation). Consequently, in case of weak system noise, the resource cost of MPDO will be significantly less than that of the MPO due to a relatively small inner dimension needed for the simulation. In case of strong system noise, a relatively small bond dimension may be sufficient to simulate the noisy circuits, indicating a regime that the noise is large enough for an `easy' classical simulation. Moreover, we propose a more effective tensor updates scheme with optimal truncations for both the inner and the bond dimensions, performed after each layer of the circuit, which enjoys a canonical form of the MPDO for improving simulation accuracy. With truncated inner dimension to a maximum value $\kappa$ and bond dimension to a maximum value $\chi$, the cost of our simulation scales as $\sim ND\kappa^3\chi^3$, for an $N$-qubit circuit with depth $D$.

Utilizing GPGPUs for simulating quantum circuit

State vector describing the state of an n-qubit quantum system requires 2 n complex numbers, which means the memory usage of simulating a quantum computer grows exponentially with respect to the number of qubits. For the memory of a single hardware is limited, distributed computation is necessary for simulating quantum systems with large number of qubits. In our proposal, we developed a multi-GPU solution. State vector is divided into several chunks and allocated to different GPU device. Making it possible to simulate quantum circuits of any size on appropriate GPU cluster. We analyse the theoretical computation resource and come to the conclusion that memory copy bandwidth between GPUs is the bottleneck of time consumption. Total execution time is O(logDN/DN) if peer-to-peer memory copy is supported, otherwise time elapsed is O(logDN) where DN is number of GPU devices. Memory consumption on one GPU is O(1/DN).

Simulating quantum circuits by adiabatic computation: Improved spectral gap bounds

Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L \gg 1$ is $\min_s\Delta \gtrsim \pi^2 / [8(L+1)^2]$, an improvement over previous estimates. Our first method is a direct approach based on an eigenstate ansatz, while the the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.

Simulation of quantum circuits by low-rank stabilizer decompositions

Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion ofstabilizerrank, which for a pure stateψis defined to be the smallest integerχsuch thatψis a superposition ofχstabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions ofχ∼106stabilizer states and sample from the fulln-bit output distribution, improving on previous simulations which used∼103stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64Tgates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.

Clifford recompilation for faster classical simulation of quantum circuits

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 factorℓ/m, whereℓandmare respectively the total and non-Clifford gate counts. The main technique is a circuit recompilation routine based on manipulating exponentiated Pauli operators. The recompilation routine also facilitates numerical search for Clifford decompositions of products of non-Clifford gates, which can further reduce the runtime in certain cases by reducing the 1-norm of the vector of expansion,‖a‖1. It may additionally lead to a framework for optimizing circuit implementations over a gate-set, reducing the overhead for state-injection in fault-tolerant implementations. We provide a concise exposition of randomized sparsification, and describe how to use it to estimate circuit amplitudes in a way which can be generalized to a broader class of gates and states. This latter method can be used to obtain additive error estimates of circuit probabilities with a faster runtime than the full techniques of Bravyi et al. Such estimates are useful for validating near-term quantum devices provided that the target probability is not exponentially small.

QTorch: The quantum tensor contraction handler.

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 on 3- through 7-regular graphs using the quantum approximate optimization algorithm (QAOA), successfully simulating up to 100 qubits. We test two different methods for generating the ordering of tensor index contractions: one is based on the tree decomposition of the line graph, while the other generates ordering using a straight-forward stochastic scheme. Through studying instances of QAOA circuits, we show the expected result that as the treewidth of the quantum circuit’s line graph decreases, TN contraction becomes significantly more efficient than simulating the whole Hilbert space. The results in this work suggest that tensor contraction methods are superior only when simulating Max-Cut/QAOA with graphs of regularities approximately five and below. Insight into this point of equal computational cost helps one determine which simulation method will be more efficient for a given quantum circuit. The stochastic contraction method outperforms the line graph based method only when the time to calculate a reasonable tree decomposition is prohibitively expensive. Finally, we release our software package, qTorch (Quantum TensOR Contraction Handler), intended for general quantum circuit simulation. For a nontrivial subset of these quantum circuits, 50 to 100 qubits can easily be simulated on a single compute node.

Generalized counting constraint satisfaction problems with determinantal circuits

Generalized counting constraint satisfaction problems include Holant problems with planarity restrictions; polynomial-time algorithms for such problems include matchgates and matchcircuits, which are based on Pfaffians. In particular, they use gates which are expressible in terms of a vector of sub-Pfaffians of a skew-symmetric matrix. We introduce a new type of circuit based instead on determinants, with seemingly different expressive power. In these determinantal circuits, a gate is represented by the vector of all minors of an arbitrary matrix. Determinantal circuits permit a different class of gates. Applications of these circuits include proofs of theorems from algebraic graph theory including the Chung–Langlands formula for the number of rooted spanning forests of a graph and computing Tutte polynomials of certain matroids. They also give a strategy for simulating quantum circuits with closed timelike curves. Monoidal category theory provides a useful language for discussing such counting problems, turning combinatorial restrictions into categorical properties. We introduce the counting problem in monoidal categories and count-preserving functors as a way to study FP subclasses of problems in settings which are generally #P-hard. Using this machinery we show that, surprisingly, determinantal circuits can be simulated by Pfaffian circuits at quadratic cost.

Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

We study the classical simulatability of constant-depth polynomial-size quantum circuits followed by only one single-qubit measurement, where the circuits consist of universal gates on at most two ...