Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient particle-conserving brick-wall quantum circuits

Babatunde M. Ayeni

TL;DR

This paper shows how to construct efficient particle-conserving gates using some practical ideas from symmetric tensor networks, including the generalized one, and shows that the general particle-conserving gate with only four real parameters is generally best.

Abstract

In variational quantum optimization with particle-conserving quantum circuits, it is often difficult to decide a priori which particle-conserving gates and circuit ansatzes would be most efficient for a given problem. This is important especially for noisy intermediate-scale quantum (NISQ) processors with limited resources. While this may be challenging to answer in general, deciding which particle-conserving gate would be most efficient is easier within a specified circuit ansatz. In this paper, we show how to construct efficient particle-conserving gates using some practical ideas from symmetric tensor networks. We derive different types of particle-conserving gates, including the generalized one. We numerically test the gates under the framework of brick-wall circuits. We show that the general particle-conserving gate with only four real parameters is generally best. In addition, we present an algorithm to extend brick-wall circuit with two-qubit nearest-neighbouring gates to non-nearest-neighbouring gates. We test and compare the efficiency of the circuits with Heisenberg spin chain with and without next-nearest-neighbouring interactions.

Efficient particle-conserving brick-wall quantum circuits

TL;DR

This paper shows how to construct efficient particle-conserving gates using some practical ideas from symmetric tensor networks, including the generalized one, and shows that the general particle-conserving gate with only four real parameters is generally best.

Abstract

In variational quantum optimization with particle-conserving quantum circuits, it is often difficult to decide a priori which particle-conserving gates and circuit ansatzes would be most efficient for a given problem. This is important especially for noisy intermediate-scale quantum (NISQ) processors with limited resources. While this may be challenging to answer in general, deciding which particle-conserving gate would be most efficient is easier within a specified circuit ansatz. In this paper, we show how to construct efficient particle-conserving gates using some practical ideas from symmetric tensor networks. We derive different types of particle-conserving gates, including the generalized one. We numerically test the gates under the framework of brick-wall circuits. We show that the general particle-conserving gate with only four real parameters is generally best. In addition, we present an algorithm to extend brick-wall circuit with two-qubit nearest-neighbouring gates to non-nearest-neighbouring gates. We test and compare the efficiency of the circuits with Heisenberg spin chain with and without next-nearest-neighbouring interactions.
Paper Structure (31 sections, 1 theorem, 52 equations, 7 figures, 2 tables)

This paper contains 31 sections, 1 theorem, 52 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generic $\mathbb{Z}_2$-symmetric and particle-conserving gates
  3. The general idea
  4. Charge-conserving operator in a symmetry basis
  5. Example: Two-qubit with $\mathbb{Z}_2$ symmetry.
  6. $\mathbb{Z}_2$-symmetric basis transformations
  7. Generic particle-conserving gates
  8. On two neighbouring qubits
  9. On multiple qubits
  10. Types of particle-conserving gate parameterizations
  11. NN gate with two parameters in charge-1 block
  12. NN gate with two parameters in charge-$1$ and charge-$2$ blocks
  13. NN two-qubit general particle-conserving gate
  14. Long-range single excitation two-qubit gates
  15. Particle-Conserving Brick-wall Circuits
  16. ...and 16 more sections

Key Result

Theorem 1

Charge-conserving operators in a symmetry basis can be realized as controlled operations, and vice versa.

Figures (7)

  • Figure 1: A brick-wall parameterized quantum circuit with "placeholder" gates $M$. We choose M from the list of gates $\{ A,B,G\}$ presented in Sec. \ref{['Sec:PC gates']}. The initial $N$ number of X gates is to bring the quantum processor into the $\mathcal{H}_{N,L}$ Fock space (as explained in the text), and then followed by an alternating sequence of odd and even gates. In general, the gates $M$ all have different parameters.
  • Figure 2: Parameterized quantum circuit for $3$ particles on $8$ sites with both NN and NNN gates. This creates an ansatz state in the Fock space $\mathcal{H}_{3,8}$. The circuit consists of three alternating layers of NN and NNN gates. The first layer consists of NN gates, the second layer consists of NNN gates, and the third layer consists of another NN gates. Note that the gates $M$s are functions of some parameters. To reduce computational cost, we will choose $M$ as either $A(\theta, \phi)$ or $B(\theta, \phi)$.
  • Figure 3: Results of the variational optimization of Heisenberg model $(a)$-$(c)$ for lattice sizes $L=4,6,8$ and its XX reduction (d) for $L=8$ sites. The horizontal axis ("opt. steps") is the number of classical optimization steps (or function evaluations), and the vertical axis ("relative error") is the error relative to the true ground state energy. In all the plots, the red "dashed-dotted" line is for the A-gate, while the blue dashed line is for the B-gate, and the green solid line is for the G-gate.
  • Figure 4: Results of variational optimization of NN and NNN Heisenberg model using parameterized quantum circuits with and without NNN gates. The vertical axis is the relative error away from exact lowest energy, while the horizontal axis is the number of function evaluations. We trial only circuits $C_A$ with only NN gate-$A$ gates---shown as the dashed lines--- and $C_A^{\mathrm{ex}}$ with both NN and NNN gates---shown as solid lines. Exact ground energy $E_0 = -14.7262$ for a lattice with $8$ sites. We considered four different circuit layers, $m=3,5,6,10$. No significant relative advantage in using circuits with additional NNN gates is seen.
  • Figure 5: Plots of average fidelity $\bar{F}_{\mathrm{sample}}$ (i.e. fidelity per trial) of learning uniform randomly sampled states in Fock space $\mathcal{H}_{N,L}$ with brick-wall circuits $C_A$, $C_B$, and $C_G$. The Fock spaces considered are (a) $\mathcal{H}_{2,4}$ (b) $\mathcal{H}_{3,5}$, and (c) $\mathcal{H}_{3,6}$. The data points are the average fidelity $\bar{F}_{\mathrm{sample}}$, averaged over $N_T = 10$ trials, and plotted against the $N_S = 250$ random samples. The lines are the means $f$ (i.e. average fidelity per sample per trial) for the corresponding data points. Overall, gate $G$ has superior learning capability than gate $A$, which in turn is also better than gate $B$.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1