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Benchmarking Quantum Simulation Methods

Calvin Ku, Yu-Cheng Chen, Alice Hu, Min-Hsiu Hsieh

TL;DR

The impact of three key parameters on the overall quantum resource costs for the QPE algorithm are quantified: the choice between trotterization and qubitization, the use of molecular orbitals versus plane-wave basis-sets, and the selection of the fermion-to-qubit encoding scheme.

Abstract

Quantum Phase Estimation (QPE) is a cornerstone algorithm for fault-tolerant quantum computation, especially for electronic structure calculations of chemical systems. To accommodate the diverse characteristics of quantum chemical systems, numerous variants of QPE have been developed, each with distinct qubit and gate cost implications. In this paper, we quantify the impact of three key parameters on the overall quantum resource costs for the QPE algorithm: the choice between trotterization and qubitization, the use of molecular orbitals versus plane-wave basis-sets, and the selection of the fermion-to-qubit encoding scheme. From this, we establish clear performance trade-offs and delineate specific parameter regimes that minimize resource costs for relevant molecular systems. When performing phase estimation on large molecules in the fault-tolerant setting, we found the first-quantized qubitization circuit using the plane-wave basis to be the most efficient, with a gate cost scaling of $\tilde{\mathcal{O}}([N^{4/3}M^{2/3}+N^{8/3}M^{1/3}]/\varepsilon)$ for a system of $N$ electrons and $M$ orbitals, which is the best known scaling to date. On the other hand, when only noisy intermediate-scale or near-term fault-tolerant systems are available, the phase estimation of small molecules can be performed with gate cost of $\mathcal{O}(M^{7}/\varepsilon^{2})$ via trotterization in the MO basis. Furthermore, we provide numerical estimations of qubit and T gate costs required to perform QPE for several real-world molecular systems under these different parameter choices.

Benchmarking Quantum Simulation Methods

TL;DR

The impact of three key parameters on the overall quantum resource costs for the QPE algorithm are quantified: the choice between trotterization and qubitization, the use of molecular orbitals versus plane-wave basis-sets, and the selection of the fermion-to-qubit encoding scheme.

Abstract

Quantum Phase Estimation (QPE) is a cornerstone algorithm for fault-tolerant quantum computation, especially for electronic structure calculations of chemical systems. To accommodate the diverse characteristics of quantum chemical systems, numerous variants of QPE have been developed, each with distinct qubit and gate cost implications. In this paper, we quantify the impact of three key parameters on the overall quantum resource costs for the QPE algorithm: the choice between trotterization and qubitization, the use of molecular orbitals versus plane-wave basis-sets, and the selection of the fermion-to-qubit encoding scheme. From this, we establish clear performance trade-offs and delineate specific parameter regimes that minimize resource costs for relevant molecular systems. When performing phase estimation on large molecules in the fault-tolerant setting, we found the first-quantized qubitization circuit using the plane-wave basis to be the most efficient, with a gate cost scaling of for a system of electrons and orbitals, which is the best known scaling to date. On the other hand, when only noisy intermediate-scale or near-term fault-tolerant systems are available, the phase estimation of small molecules can be performed with gate cost of via trotterization in the MO basis. Furthermore, we provide numerical estimations of qubit and T gate costs required to perform QPE for several real-world molecular systems under these different parameter choices.

Paper Structure

This paper contains 21 sections, 3 theorems, 74 equations, 24 figures, 18 tables.

Table of Contents

  1. Details on Encoding Schemes
  2. First-Quantized Encoding
  3. Sorted-List Encoding
  4. Additional Optimizations
  5. Hamiltonian Simulation
  6. Robust Phase Estimation with Trotterization
  7. Trotterization for the Plane-Wave Basis
  8. Szegedy Quantum Walk with Qubitization
  9. Single-Factorization
  10. First-Quantized Encoding
  11. Trotterization Implementation for the Sorted-List Encoding
  12. Circuit Implementations
  13. Cost Analysis
  14. Simulation Methods for Trotterization cost
  15. Qubitization Implementation for the Sorted-List Encoding
  16. ...and 6 more sections

Key Result

Theorem 1

The ground-state energy of a Hamiltonian $H$ can be calculated to precision $\varepsilon$ using the RPE algorithm. For the sorted-list encoding, the gate costs (both Clifford and T gates) scale as: For the partially random approach, $L_{\text{det}}$ and $\lambda_{\text{rand}}$ only refer to the deterministic and random parts, respectively. The qubit costs scales identically across these three str

Figures (24)

  • Figure 1: Workflow for an energy estimation algorithm using quantum computers, using either trotterization or qubitization to effectively implement the quantum chemical Hamiltonian into the quantum circuit.
  • Figure A.1: The $=p$ circuit carolanSuccinctFermionData2024. This circuit flips ancilla $\ket{a}$ when the value of $i$ is equal to (\ref{['fig:eq_const_circuit']}) a constant $p$ or (\ref{['fig:eq_index_circuit']}) an indexed value $p$.
  • Figure A.2: The $<p$ circuit carolanSuccinctFermionData2024. This circuit flips ancilla $\ket{a}$ when the value of $i$ is less than (\ref{['fig:lt_const_circuit']}) a constant $p$ or (\ref{['fig:lt_index_circuit']}) an indexed value $p$. While our work only uses the $<p$ gate, the original work includes the $\le p$ and $>p$ variant which can be implemented similarly berryQubitizationArbitraryBasis2019.
  • Figure A.3: The bubble circuit $U_p$carolanSuccinctFermionData2024. This circuit swaps the two registers $a$ and $b$ when one of them is equal to $p$ and the other is larger than $p$.
  • Figure A.4: The $a\leftrightarrow b$ circuit carolanSuccinctFermionData2024. This circuit outputs $b$ when the input is $a$ and outputs $a$ when the input is $b$. Otherwise, the input is unchanged. The $X^{a\oplus b}$ is composed of multiple CNOT gate according to the binary representation of $a\oplus b$.
  • ...and 19 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem 1
  • proof : Proof of Theorem 2
  • proof : Proof of Theorem 3