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CaRBM: A Fixed-Depth Quantum Algorithm with Partial Correction for Thermal State Preparation

Omar Alsheikh, A. F. Kemper, Ermal Rrapaj, Goksu C. Toga

Abstract

We introduce the CaRBM algorithm for fixed-depth thermal state preparation. Our algorithm is based on thermal state purification and uses the Restricted Boltzmann Machine (RBM) block-encoding scheme to implement the imaginary-time propagator $e^{-βH}$, which is implemented in the quantum circuit in a fixed-depth manner via Cartan decomposition. Our algorithm performs best at high temperatures, with the success probability of the block encoding decreasing as the temperature decreases. To increase the success probability, we have devised a correction scheme for the block-encoding that increases the temperature range our algorithm reliably probes. We demonstrate our algorithm by calculating the partition function zeros of the XXZ model and the phase diagram of the Gross-Neveu model, which is a model of strongly interacting relativistic fermions.

CaRBM: A Fixed-Depth Quantum Algorithm with Partial Correction for Thermal State Preparation

Abstract

We introduce the CaRBM algorithm for fixed-depth thermal state preparation. Our algorithm is based on thermal state purification and uses the Restricted Boltzmann Machine (RBM) block-encoding scheme to implement the imaginary-time propagator , which is implemented in the quantum circuit in a fixed-depth manner via Cartan decomposition. Our algorithm performs best at high temperatures, with the success probability of the block encoding decreasing as the temperature decreases. To increase the success probability, we have devised a correction scheme for the block-encoding that increases the temperature range our algorithm reliably probes. We demonstrate our algorithm by calculating the partition function zeros of the XXZ model and the phase diagram of the Gross-Neveu model, which is a model of strongly interacting relativistic fermions.
Paper Structure (12 sections, 9 theorems, 41 equations, 5 figures)

This paper contains 12 sections, 9 theorems, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Cartan decomposition
  4. Finite-temperature block encoding
  5. Correction scheme for the improving sampling rate
  6. Applications
  7. Partition function zeros
  8. Thermal phase diagram of relativistic fermions in $1+1d$
  9. Conclusion and Outlook
  10. Block-encoding parameters
  11. Layer Corrections
  12. Gross-Neveu Model

Key Result

Theorem 1

For the ITE process $e^{-\kappa_j\sigma^j}e^{-\kappa_{j-1}\sigma^{j-1}}\dots e^{-\kappa_1\sigma^1}|\psi\rangle$, we can find an operator $O$ that anticommutes with $\sigma^j$ but commutes with all the other layers, provided that $\sigma^j$ cannot be written as a product of the other Pauli strings. T

Figures (5)

  • Figure 1: (a) The three building blocks of CaRBM are shown. First, the Cartan decomposition is used to transform the imaginary time propagator $e^{-\beta H}$ into a product of commuting propagators. These commuting propagators are then encoded via RBM into unitary operations by coupling the system to an ancilla. The first few layers can be corrected by adding a controlled operator on the ancilla, ensuring a perfect success rate, while the subsequent layers are encoded probabilistically by post-selecting on the ancilla. (b) The full circuit for $e^{-\beta H}$ using CaRBM.
  • Figure 2: Schematics for the different circuits used for (a) infinite-temperature thermal state preparation, (b) Lee-Yang zeros, (c) Fisher zeros, and (d) thermal averages.
  • Figure 3: Top: Circuit simulations of partition function plots showing the locations of Lee-Yang Zeros for the 4-site XXZ model coupled to a complexified probe field. The behavior of the zeros undergoes a considerable change as we increase the value of the ratio $J/|J_z|$, signifying a phase transition from the Ising-like phase to the XY-like phase. Bottom: Line plots of sections taken on the heat maps. The first two are taken at $\mathrm{Re}(g) = 0$ and the second two are taken at $\mathrm{Im}(g) = \pi/2$. The sections are chosen to pass through the zeros. The markers are the circuit simulation results, and the shaded regions are confidence intervals at a confidence level of 0.95, estimated via bootstrap. The solid lines are MPS calculations. All simulations are at $\beta = 1$.
  • Figure 4: Complex partition function showing the locations of Fisher Zeros for 4-site XXZ model obtained via simulations of the circuits discussed in the main text.
  • Figure 5: Top: Circuit simulations $\sum_i \langle Z_iZ_0\rangle$ for the 2-site Gross-Neveu model with 2 fermion flavors across a range of temperatures $\beta$ and chemical potentials $\mu$. Simulations without and with correction layers are shown for comparison. Bottom: Success rate of the post-selection process. The correction procedure allows us to explore an extended region in the phase space that was otherwise inaccessible (the blackened region) due to vanishing success probabilities.

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1
  • Definition 1: Binary Symplectic Representation
  • Remark 1
  • Remark 2
  • Lemma 2
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • ...and 2 more