arXiv:1802.05834

On discrete Wigner transforms

Zhenning Cai, Jianfeng Lu, Kevin D. Stubbs

Published Feb 16, 2018• 2 citations

Abstract

In this work, we derive a discrete analog of the Wigner transform over the space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer $N$. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for $p=2$ or more generally of the Heisenberg-Weyl group elements for $p > 2$. We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all $p$, Wootters' construction corresponds to taking the inverse symplectic Fourier transform instead of the inverse Fourier transform. Finally, we discuss some implications of these results for the numerical simulation of many-body quantum spin systems.

DOI: 10.48550/arxiv.1802.05834

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arXiv:1802.05834

On discrete Wigner transforms

Zhenning Cai, Jianfeng Lu, Kevin D. Stubbs

Published Feb 16, 2018• 2 citations

arXiv HTML PDF

Abstract

In this work, we derive a discrete analog of the Wigner transform over the space

(\mathbb{C}^p)^{\otimes N}

for any prime

p

and any positive integer

N

. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for

p=2

or more generally of the Heisenberg-Weyl group elements for

p > 2

. We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all

p

, Wootters' construction corresponds to taking the inverse symplectic Fourier transform instead of the inverse Fourier transform. Finally, we discuss some implications of these results for the numerical simulation of many-body quantum spin systems.

DOI: 10.48550/arxiv.1802.05834

Related papers

The Heisenberg-Weyl-parity group its coherent states and a unified Wigner-Weyl function

A. Vourdas

The Heisenberg-Weyl group $HW(d)$ related to a $d$ -dimensional Hilbert space $H(d)$ , is enlarged into the Heisenberg-Weyl-parity group $HWP(d)$ that incorporates parity transformations. It consists of $2d^3$ elements, of which $d^3$ elements belong to the $HW(d)$ subgroup, and extra $d^3$ elements which are related through a Fourier transform with the former ones. It is shown that $HWP(d)$ is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. $HWP(d)$ is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. $2d^2$ coherent states related to the $HWP(d)$ group are introduced, which consist of $d^2$ coherent states related to the $HW(d)$ subgroup, and extra $d^2$ coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the $2d^2$ coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the $d^2$ coherent states related to $HW(d)$ . One of the consequences of the $HWP(d)$ group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.The Heisenberg-Weyl group $HW(d)$ related to a $d$-dimensional Hilbert space $H(d)$, is enlarged into the Heisenberg-Weyl-parity group $HWP(d)$ that incorporates parity transformations. It consists of $2d^3$ elements, of which $d^3$ elements belong to the $HW(d)$ subgroup, and extra $d^3$ elements which are related through a Fourier transform with the former ones. It is shown that $HWP(d)$ is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. $HWP(d)$ is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.$2d^2$ coherent states related to the $HWP(d)$ group are introduced, which consist of $d^2$ coherent states related to the $HW(d)$ subgroup, and extra $d^2$ coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the $2d^2$ coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the $d^2$ coherent states related to $HW(d)$. One of the consequences of the $HWP(d)$ group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.

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The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.

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Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems

Chao Lei, Apostol Vourdas

Abstract Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=d^n$ D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space $D=d_0...d_{n-1}$ D = d 0 . . . d n - 1 with $d_0,...,d_{n-1}$ d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from $\mathcal{O}(D^2)$ O ( D 2 ) to $\mathcal{O}(D\log D)$ O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.Abstract Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $$D=d^n$$ D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space $$D=d_0...d_{n-1}$$ D = d 0 . . . d n - 1 with $$d_0,...,d_{n-1}$$ d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from $$\mathcal{O}(D^2)$$ O ( D 2 ) to $$\mathcal{O}(D\log D)$$ O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

arXiv:2405.05163The European Physical Journal PlusMay 8, 20240 citations

Complete Set of Unitary Irreps of Discrete Heisenberg Group $HW_{2^{s}}$ Complete Set of Unitary Irreps of Discrete Heisenberg Group $$ HW_{2^{s}}$$

Emmanuel G. Floratos, I. Tsohantjis

Abstract Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group $HW_{2^{s}}$ H W 2 s over the discrete phase space lattice $\mathbb {Z}_{2^{s}}\otimes$ Z 2 s ⊗ $\mathbb {Z}_{2^{s}}$ Z 2 s is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.Abstract Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group $$HW_{2^{s}}$$ H W 2 s over the discrete phase space lattice $$\mathbb {Z}_{2^{s}}\otimes $$ Z 2 s ⊗ $$\mathbb {Z}_{2^{s}}$$ Z 2 s is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.

arXiv:2210.04263International Journal of Theoretical PhysicsJan 18, 20251 citation

In this work, we derive a discrete analog of the Wigner transform over the space

(\mathbb{C}^p)^{\otimes N}

for any prime

p

and any positive integer

N

. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for

p=2

or more generally of the Heisenberg-Weyl group elements for

p > 2

. We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all

p

The Heisenberg-Weyl group

HW(d)

related to a

d

-dimensional Hilbert space

H(d)

, is enlarged into the Heisenberg-Weyl-parity group

HWP(d)

that incorporates parity transformations. It consists of

2d^3

elements, of which

d^3

elements belong to the

HW(d)

subgroup, and extra

d^3

elements which are related through a Fourier transform with the former ones. It is shown that

HWP(d)

is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed.

HWP(d)

is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.

2d^2

coherent states related to the

HWP(d)

group are introduced, which consist of

d^2

coherent states related to the

HW(d)

subgroup, and extra

d^2

coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the

2d^2

coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the

d^2

coherent states related to

HW(d)

. One of the consequences of the

HWP(d)

group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.

We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality

N

. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator

P

and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.

Abstract Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space

D=d^n

D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space

D=d_0...d_{n-1}

D = d 0 . . . d n - 1 with

d_0,...,d_{n-1}

d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from

\mathcal{O}(D^2)

O ( D 2 ) to

\mathcal{O}(D\log D)

O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

Complete Set of Unitary Irreps of Discrete Heisenberg Group

HW_{2^{s}}

Abstract Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group

HW_{2^{s}}

H W 2 s over the discrete phase space lattice

\mathbb {Z}_{2^{s}}\otimes

Z 2 s ⊗

\mathbb {Z}_{2^{s}}

Z 2 s is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.