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Weyl's Relations, Integrable Matrix Models and Quantum Computation

B. Sriram Shastry, Emil A. Yuzbashyan, Aniket Patra

Abstract

Starting from a generalization of Weyl's relations in finite dimension $N$, we show that the Heisenberg commutation relations can be satisfied in a specific $N-1$ dimensional subspace, and display a linear map for projecting operators to this subspace. This setup is used to construct a hierarchy of parameter-dependent commuting matrices in $N$ dimensions. This family of commuting matrices is then related to Type-1 matrices representing quantum integrable models. The commuting matrices find an interesting application in quantum computation, specifically in Grover's database search problem. Each member of the hierarchy serves as a candidate Hamiltonian for quantum adiabatic evolution and, in some cases, achieves higher fidelity than standard choices -- thus offering improved performance.

Weyl's Relations, Integrable Matrix Models and Quantum Computation

Abstract

Starting from a generalization of Weyl's relations in finite dimension , we show that the Heisenberg commutation relations can be satisfied in a specific dimensional subspace, and display a linear map for projecting operators to this subspace. This setup is used to construct a hierarchy of parameter-dependent commuting matrices in dimensions. This family of commuting matrices is then related to Type-1 matrices representing quantum integrable models. The commuting matrices find an interesting application in quantum computation, specifically in Grover's database search problem. Each member of the hierarchy serves as a candidate Hamiltonian for quantum adiabatic evolution and, in some cases, achieves higher fidelity than standard choices -- thus offering improved performance.

Paper Structure

This paper contains 13 sections, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weyl matrices and the limit $N\to \infty$
  3. Weyl matrices with N finite
  4. The algebra of Type-1 matrices and connection with generalized Weyl matrices
  5. A linear map on operators
  6. Analogy with stochastic equations
  7. The parameter-dependent matrix Hamiltonian and its conservation laws
  8. Mutual Commutation of $I_n$
  9. Proof of cancellation of Eq. (\ref{['kscommute']}):
  10. Spectra, recurrence relation, general formula, and $I_n$ as a polynomial in $H$:
  11. Quantum Computation with Type-1 Matrices
  12. Concluding Remarks
  13. Deriving the Heisenberg Algebra from the Weyl Matrices by Taking the $N\to \infty$ Limit

Figures (2)

  • Figure 1: Efficiency increase of the Grover's search algorithm by utilizing quantum integrability of Type-1 matrices. Panel (a): Instantaneous energy gaps between the ground and first excited state as functions of $t\in (0, T_\mathrm{run})$ for two random sets of $\epsilon_{i>2}$ with $\Delta\epsilon = 0.1$ and three different currents $I_{n}$ as well as for the Grover Hamiltonian (solid black line). Panel (b): $(1 - F_{n})$ for different $I_n(t)$ and the same random sets of $\epsilon_{i>2}$ as in panel (a), where $F_n$ is the fidelity. The increased fidelity as compared to the Grover Hamiltonian (horizontal black line) is due to quantum interference. This effect is particularly dramatic for $n=3$ and $n=7$ with $1-F_3 = 5.20 \times 10^{-6}$ and $1-F_7 = 6.40 \times 10^{-5}$ for the random choice #2 RandSeed. The matrix size is $N = 64$ and the parameter $\delta = 0.1$ in the interpolating function $s(t)$.
  • Figure 2: Efficiency gain using $I_3$ for matrix size $N = 64$. Panel (a): Five values of $\Delta\epsilon$, spaced logarithmically, are tested with fixed $\delta = 0.1$. Random choices $\#1$ and $\#2$ are shown as blue crosses and red squares, respectively RandSeed. As $\Delta\epsilon \to 0$, fidelity-deficit converges to that of the Grover Hamiltonian (black line). Notably, the performance of $I_3$ improves markedly at $\Delta\epsilon = 0.1$. Panel (b): Holding $\Delta\epsilon = 0.1$ fixed, we vary $\delta$ using the same random $\epsilon_i$ as in Fig. \ref{['Fig:Infid_Gaps_vs_n']}. As expected, both $T_\mathrm{run}$ and $F_3$ rise with decreasing $\delta$. Curiously, lowering $\delta$ brings out a marked efficiency gain in random choice $\#1$, driven by quantum integrability.