We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors in the operator converge to a common tensor as the number of qubits in the transform increases. Together these results imply that the application of the quantum Fourier transform to a matrix product state with $n$ qubits of maximum Schmidt rank $χ$ can be simulated in $O(n (log(n))^2 χ^2)$ time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors in the operator converge to a common tensor as the number of qubits in the transform increases. Together these results imply that the application of the quantum Fourier transform to a matrix product state with $n$ qubits of maximum Schmidt rank $χ$ can be simulated in $O(n (log(n))^2 χ^2)$ time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.

arXiv:1406.0931Jun 4, 20141 citation

Quantum Fourier Transform Has Small Entanglement

Jielun Chen, Edwin Miles Stoudenmire, Steven R. White

The quantum Fourier transform (QFT) is a key component of many important quantum algorithms, most famously being the essential ingredient in Shor's algorithm for factoring products of primes.Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically.While early results showed the QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT.The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can generate only a constant amount of entanglement regardless of the number of qubits.In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively.Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform on many classes of functions.We demonstrate this speedup in test calculations on some simple functions.For data vectors of length 10 6 -10 8 , the speedup can be a few orders of magnitude.The quantum Fourier transform (QFT) is a key component of many important quantum algorithms, most famously being the essential ingredient in Shor's algorithm for factoring products of primes.Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically.While early results showed the QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT.The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can generate only a constant amount of entanglement regardless of the number of qubits.In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively.Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform on many classes of functions.We demonstrate this speedup in test calculations on some simple functions.For data vectors of length 10 6 -10 8 , the speedup can be a few orders of magnitude.

arXiv:2210.08468PRX QuantumOct 27, 202363 citations

A parallel environment for simulating quantum computation

Geva Patz

This thesis describes the design and implementation of an environment to allow quantum computation to be simulated on classical computers. Although it is believed that quantum computers cannot in general be efficiently simulated classically, it is nevertheless possible to simulate small but interesting systems, on the order of a few tens of quantum bits. Since the state of the art of physical implementations is less than 10 bits, simulation remains a useful tool for understanding the behavior of quantum algorithms. To create a suitable envrionment for simulation, we constructed a 32-node cluster of workstation class computers linked with a high speed (gigabit Ethernet) network. We then wrote an initial simulation environment based on parallel linear algebra libraries with a Matlab front end. These libraries operated on large matrices representing the problem being simulated. The parallel Matlab environment demonstrated a degree of parallel speedup as we added processors, but overall execution times were high, since the amount of data scaled exponentially with the size of the problem. This increased both the number of operations that had to be performed to compute the simulation, and the volume of data that had to be communicated between the nodes as they were computing. The scaling also affected memory utilization, limiting us to a maximum problem size of 14 qubits. In an attempt to increase simulation efficiency, we revisited the design of the simulation environment. Many quantum algorithms have a structure that can be described using the tensor product operator from linear algebra. We believed that a new simulation environment based on this tensor product structure would be substantially more efficient than one based on large matrices. We designed a new simulation envrionment that exploited this tensor product structure. Benchmarks that we performed on the new simulation environment confirmed that it was substantially more efficient, allowing us to perform simulations of the quantum Fourier transform and the discrete approximation to the solution of 3-SAT by adiabatic evolution up to 25 qubits in a reasonable time. Thesis Supervisor: Stephen A. Benton Title: Professor of Media Arts and SciencesThis thesis describes the design and implementation of an environment to allow quantum computation to be simulated on classical computers. Although it is believed that quantum computers cannot in general be efficiently simulated classically, it is nevertheless possible to simulate small but interesting systems, on the order of a few tens of quantum bits. Since the state of the art of physical implementations is less than 10 bits, simulation remains a useful tool for understanding the behavior of quantum algorithms. To create a suitable envrionment for simulation, we constructed a 32-node cluster of workstation class computers linked with a high speed (gigabit Ethernet) network. We then wrote an initial simulation environment based on parallel linear algebra libraries with a Matlab front end. These libraries operated on large matrices representing the problem being simulated. The parallel Matlab environment demonstrated a degree of parallel speedup as we added processors, but overall execution times were high, since the amount of data scaled exponentially with the size of the problem. This increased both the number of operations that had to be performed to compute the simulation, and the volume of data that had to be communicated between the nodes as they were computing. The scaling also affected memory utilization, limiting us to a maximum problem size of 14 qubits. In an attempt to increase simulation efficiency, we revisited the design of the simulation environment. Many quantum algorithms have a structure that can be described using the tensor product operator from linear algebra. We believed that a new simulation environment based on this tensor product structure would be substantially more efficient than one based on large matrices. We designed a new simulation envrionment that exploited this tensor product structure. Benchmarks that we performed on the new simulation environment confirmed that it was substantially more efficient, allowing us to perform simulations of the quantum Fourier transform and the discrete approximation to the solution of 3-SAT by adiabatic evolution up to 25 qubits in a reasonable time. Thesis Supervisor: Stephen A. Benton Title: Professor of Media Arts and Sciences

Jan 1, 20036 citations

Efficient classical simulation of the approximate quantum Fourier transform

Nadav Yoran, Anthony J. Short

We present a method for classically simulating quantum circuits based on the tensor contraction model of Markov and Shi (e-print arXiv:quant-ph/0511069). Using this method we are able to classically simulate the approximate quantum Fourier transform in polynomial time. Moreover, our approach allow 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 present a method for classically simulating quantum circuits based on the tensor contraction model of Markov and Shi (e-print arXiv:quant-ph/0511069). Using this method we are able to classically simulate the approximate quantum Fourier transform in polynomial time. Moreover, our approach allow 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.

arXiv:quant-ph/0611241Physical Review AOct 16, 200755 citations

Quantum Fourier transform revisited

Camps, D, Van Beeumen, R, Yang, C

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

Jan 1, 202125 citations

Classical Simulability from Operator Entanglement Scaling

Neil Dowling

Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE $α$ -Rényi entropies. Specifically, we prove that a volume law scaling for $α\geq 1$ implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the $α< 1$ entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE $α$-Rényi entropies. Specifically, we prove that a volume law scaling for $α\geq 1$ implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the $α< 1$ entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.

quant-pharXiv:2603.05656Physical Review AMar 9, 20260 citations

n

qubits of maximum Schmidt rank

χ

can be simulated in

O(n (log(n))^2 χ^2)

time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.

α

-Rényi entropies. Specifically, we prove that a volume law scaling for

α\geq 1

implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the

α< 1

entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.