Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Efficient Quantum Circuit for Block Encoding a Pairing Hamiltonian

Diyi Liu, Weijie Du, Lin Lin, James P. Vary, Chao Yang

TL;DR

This work introduces an explicit block-encoding circuit for the nuclear pairing Hamiltonian $\mathcal{H}_{\rm pair}$ that encodes sparsity via an $O_C$ oracle built from multi-qubit controlled swaps and encodes nonzeros via an $O_{\mathcal{H}}$ oracle, avoiding the usual fermion-to-Pauli mappings. By combining these blocks with quantum signal processing, the authors construct efficient quantum circuits for DOS estimation and demonstrate the approach on toy and three-nucleon pairing Hamiltonians, achieving a polynomial-scaling gate complexity in the basis size. The method offers a direct alternative to LCU-based block-encodings and extends naturally to more general second-quantized Hamiltonians, with practical resource estimates provided for system sizes up to modest basis counts. Overall, the paper advances practical quantum-input models for second-quantized many-body problems and paves the way for scalable DOS computations on quantum hardware.

Abstract

We present an efficient quantum circuit for block encoding pairing Hamiltonian often studied in nuclear physics. Our block encoding scheme does not require mapping the creation and annihilation operators to the Pauli operators and representing the Hamiltonian as a linear combination of unitaries. Instead, we show how to encode the Hamiltonian directly using controlled swap operations. We analyze the gate complexity of the block encoding circuit and show that it scales polynomially with respect to the number of qubits required to represent a quantum state associated with the pairing Hamiltonian. We also show how the block encoding circuit can be combined with the quantum singular value transformation to construct an efficient quantum circuit for approximating the density of states of a pairing Hamiltonian. The techniques presented can be extended to encode more general second-quantized Hamiltonians.

An Efficient Quantum Circuit for Block Encoding a Pairing Hamiltonian

TL;DR

This work introduces an explicit block-encoding circuit for the nuclear pairing Hamiltonian that encodes sparsity via an oracle built from multi-qubit controlled swaps and encodes nonzeros via an oracle, avoiding the usual fermion-to-Pauli mappings. By combining these blocks with quantum signal processing, the authors construct efficient quantum circuits for DOS estimation and demonstrate the approach on toy and three-nucleon pairing Hamiltonians, achieving a polynomial-scaling gate complexity in the basis size. The method offers a direct alternative to LCU-based block-encodings and extends naturally to more general second-quantized Hamiltonians, with practical resource estimates provided for system sizes up to modest basis counts. Overall, the paper advances practical quantum-input models for second-quantized many-body problems and paves the way for scalable DOS computations on quantum hardware.

Abstract

We present an efficient quantum circuit for block encoding pairing Hamiltonian often studied in nuclear physics. Our block encoding scheme does not require mapping the creation and annihilation operators to the Pauli operators and representing the Hamiltonian as a linear combination of unitaries. Instead, we show how to encode the Hamiltonian directly using controlled swap operations. We analyze the gate complexity of the block encoding circuit and show that it scales polynomially with respect to the number of qubits required to represent a quantum state associated with the pairing Hamiltonian. We also show how the block encoding circuit can be combined with the quantum singular value transformation to construct an efficient quantum circuit for approximating the density of states of a pairing Hamiltonian. The techniques presented can be extended to encode more general second-quantized Hamiltonians.
Paper Structure (18 sections, 3 theorems, 50 equations, 18 figures, 2 tables)

This paper contains 18 sections, 3 theorems, 50 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Block encoding and quantum signal processing
  4. Block encoding circuit
  5. The $O_C$ circuit
  6. Define $c(j,\ell)$ by bit swap
  7. The general structure of the $O_C$ circuit
  8. The circuit for $U_{\ell}$
  9. The $O_\mathcal{H}$ circuit
  10. The complete circuit
  11. Gate complexity
  12. Numerical examples
  13. $O_C$ for a simple example
  14. The three-nucleon system with the pairing Hamiltonian
  15. Construction of $O_C$ oracle
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let where $|t|\leq 1$. There exists a set of phase angles $\Phi_d \equiv \{\phi_0,,...,\phi_d\} \in \mathbb{R}^{d+1}$so that if and only if $p(t)$ and $q(t)$ are complex valued polynomials in $t$ and satisfy When $d=0$, $\deg(q)\le-1$ should be interpreted as $q=0$.

Figures (18)

  • Figure 1: Illustration of the circuit convention. The circuit is prepared with Quantikz Package kay2018tutorial.
  • Figure 2: Illustration of Controlled-Not Gates.
  • Figure 3: Illustration: the schematic circuit design of quantum singular value transformation (for an odd $d$; for an even $d$ the last $U_A$ is replaced by $U_A^{\dag}$). The additional Hadamard gate selects only the real part of the polynomial $p$.
  • Figure 4: Illustration of the circuit for matrix block encoding.
  • Figure 5: The basic structure of the $O_C$ circuit. Qubits labeled by "Other ancilla qubits" consist of a controlling qubit and a rotation qubit as discussed in sections \ref{['sec:circuit_for_U']} and \ref{['sec:Oh']}.
  • ...and 13 more figures

Theorems & Definitions (5)

  • Definition 3.1: Block encoding camps2022explicitlin2022lecture
  • Theorem 3.1: Quantum signal processing
  • Theorem 4.1
  • Theorem 4.2
  • proof