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Dianzi xuebao

A Fast Quantum Simulation Algorithm Based on State Vector

Zhiqiang Li

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We introduce symbolic representation for quantum circuit,and show how n-qubit circuit can be described as a control qubit symbolic vector and a lower-order controlled gate,instead of the representation of higher-order unitary matrix.We extend the loopless Gray code binary generation algorithm and propose n elements k position binary generation algorithm.Besides,a fast quantum simulation algorithm(FQSA) is presented,it divides a qubit state vector into several operand groups,each group is multiplied by the same unitary matrix,and then the output can be given quickly.In contrast to any other state vector based algorithm,FQSA requires less memory,and has an optimal time complexity.The experimental result of QFT indicates that FQSA is more time-efficient and it can provide more qubits circuit simulation under the same condition.

Related papers

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Quantum circuits are divided into two classes,and unitary operator expressions for two kinds of circuits in each class are offered.Transformation of unitary operators between two classes is presented according to tensor product transposition similarity theory.Besides,direct sum operation of matrices is adopted and expanded to vectors to elegantly represent the functions of quantum circuits on input vectors.On the basis of abstracting quantum circuits as controlled unitary operations,divide and conquer algorithms are proposed to simulate quantum circuits.In contrast to other algorithms based on state vector,our algorithms avoid generating matrix operators by tensor product,thus are considerably economy of memory space;and we also show that they are more time-efficient while simulating nontrivial quantum circuits.

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Abstract The simulation of quantum circuits on classical computers is an important problem in quantum computing. Such simulation requires representations of distributions over very large sets of basis vectors, and recent work has used symbolic data-structures such as Binary Decision Diagrams (BDDs) for this purpose. In this tool paper, we present Quasimodo, an extensible, open-source Python library for symbolic simulation of quantum circuits. Quasimodo is specifically designed for easy extensibility to other backends. Quasimodo allows simulations of quantum circuits, checking properties of the outputs of quantum circuits, and debugging quantum circuits. It also allows the user to choose from among several symbolic data-structures—both unweighted and weighted BDDs, and a recent structure called Context-Free-Language Ordered Binary Decision Diagrams (CFLOBDDs)—and can be easily extended to support other symbolic data-structures.

arXiv:2302.04349•Lecture notes in computer science•Jan 1, 2023•21 citations
We introduce symbolic representation for quantum circuit,and show how n-qubit circuit can be described as a control qubit symbolic vector and a lower-order controlled gate,instead of the representation of higher-order unitary matrix.We extend the loopless Gray code binary generation algorithm and propose n elements k position binary generation algorithm.Besides,a fast quantum simulation algorithm(FQSA) is presented,it divides a qubit state vector into several operand groups,each group is multiplied by the same unitary matrix,and then the output can be given quickly.In contrast to any other state vector based algorithm,FQSA requires less memory,and has an optimal time complexity.The experimental result of QFT indicates that FQSA is more time-efficient and it can provide more qubits circuit simulation under the same condition.
Quick Quantum Circuit Simulation (QQCS) is a software system for computing the result of a quantum circuit using a notation that derives directly from the circuit, expressed in a single input line. Quantum circuits begin with an initial quantum state of one or more qubits, which are the quantum analog to classical bits. The initial state is modified by a sequence of quantum gates, quantum machine language instructions, to get the final state. Measurements are made of the final state and displayed as a classical binary result. Measurements are postponed to the end of the circuit because a quantum state collapses when measured and produces probabilistic results, a consequence of quantum uncertainty. A circuit may be run many times on a quantum computer to refine the probabilistic result. Mathematically, quantum states are 2n -dimensional vectors over the complex number field, where n is the number of qubits. A gate is a 2n ×2n unitary matrix of complex values. Matrix multiplication models the application of a gate to a quantum state. QQCS is a mathematical rendering of each step of a quantum algorithm represented as a circuit, and as such, can present a trace of the quantum state of the circuit after each gate, compute gate equivalents for each circuit step, and perform measurements at any point in the circuit without state collapse. Output displays are in vector coefficients or Dirac bra-ket notation. It is an easy-to-use educational tool for students new to quantum computing.
Quantum circuits are divided into two classes,and unitary operator expressions for two kinds of circuits in each class are offered.Transformation of unitary operators between two classes is presented according to tensor product transposition similarity theory.Besides,direct sum operation of matrices is adopted and expanded to vectors to elegantly represent the functions of quantum circuits on input vectors.On the basis of abstracting quantum circuits as controlled unitary operations,divide and conquer algorithms are proposed to simulate quantum circuits.In contrast to other algorithms based on state vector,our algorithms avoid generating matrix operators by tensor product,thus are considerably economy of memory space;and we also show that they are more time-efficient while simulating nontrivial quantum circuits.
As the field of quantum computing grows, novel algorithms which take advantage of quantum phenomena need to be developed. As we are currently in the NISQ (noisy intermediate scale quantum) era, quantum algorithm researchers cannot reliably test their algorithms on real quantum hardware, which is still too limited. Instead, quantum computing simulators on classical computing systems are used. In the quantum circuit model, quantum bits (qubits) are operated on by quantum gates. A quantum circuit is a sequence of such quantum gates operating on some number of qubits. A quantum gate applied to a qubit can be controlled by other qubits in the circuit. This applies the gate only to the states which satisfy the required control qubit state. We particularly target FPGAs as our main simulation platform, as these offer potential energy savings when compared to running simulations on CPUs/GPUs. In this work, we present a memory access pattern to optimise the number of iterations that need to be scheduled to execute a quantum gate such that only the iterations which access the required pairs (determined according to the control qubits imposed on the gate) are scheduled. We show that this approach results in a significant reduction in the time required to simulate a gate for each added control qubit. We also show that this approach benefits the simulation time on FPGAs more than CPUs and GPUs and allows to outperform both CPU and GPU platforms in terms of energy efficiency, which is the main factor for scalability of the simulations.
Recently quantum computing has achieved significant results both in theoretical and experimental areas and quantum algorithms have drawn the attention of many scientists. At the present there exist several experiments that realized some parts of certain important quantum algorithms. While those experiments work with only up to seven qubits, quantum simulators allow scientists to design algorithms with many more qubits. The existing simulators have one or more of the following restrictions: either they require powerful computers and additional computation packages such as Matlab or Mathematica or they don't provide an easy method for interacting between the classical and quantum parts of an algorithm. More importantly, none of them can simulate a generic 64-qubit circuit. It is important at this stage of quantum computing development to investigate technological improvements in simulating quantum circuit design. In this thesis we developed a specialized data structure, QSV, for quantum simulators along with new method for calculation of the new state vector after application of a quantum gate in a quantum simulator. We also developed a technique that speeds up the simulation time with our QSV vectors. With those results we built a quantum simulator that overcomes many shortcomings in other simulators and it has many advanced features. Our simulator can simulate many 64-qubit circuits and it can directly simulate multiple qubit gates as well as allows users to define gates with up to 10 qubits. The simulation time is much less in comparison with other simulators we tested. The simulator defines specifications so that users can easily design and test quantum circuits. With the simulator we also developed some new quantum circuits and verified many designs and constructions.
Optimal implementation of quantum gates is crucial for designing a quantum computer. We consider the matrix representation of an arbitrary multiqubit gate. By ordering the basis vectors using the Gray code, we construct the quantum circuit which is optimal in the sense of fully controlled single-qubit gates and yet is equivalent with the multiqubit gate. In the second step of the optimization, superfluous control bits are eliminated, which eventually results in a smaller total number of the elementary gates. In our scheme the number of controlled NOT gates is O(4(n)) which coincides with the theoretical lower bound.
Abstract The simulation of quantum circuits on classical computers is an important problem in quantum computing. Such simulation requires representations of distributions over very large sets of basis vectors, and recent work has used symbolic data-structures such as Binary Decision Diagrams (BDDs) for this purpose. In this tool paper, we present Quasimodo, an extensible, open-source Python library for symbolic simulation of quantum circuits. Quasimodo is specifically designed for easy extensibility to other backends. Quasimodo allows simulations of quantum circuits, checking properties of the outputs of quantum circuits, and debugging quantum circuits. It also allows the user to choose from among several symbolic data-structures—both unweighted and weighted BDDs, and a recent structure called Context-Free-Language Ordered Binary Decision Diagrams (CFLOBDDs)—and can be easily extended to support other symbolic data-structures.