Quantum Circuit Block-Encoding for Pauli-Breit Eigenvalue Estimation
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Solution Overview
Problem
Existing quantum computing technologies face challenges in efficiently simulating and estimating the eigenvalues of Pauli-Breit Hamiltonians, which describe relativistic effects in quantum systems, due to high quantum T-gate complexity and resource inefficiencies in current quantum circuit designs.
Innovation Solution
A quantum circuit is developed that utilizes spin-mixing swap networks and orbital major mapping, combined with Majorana representation and Givens rotations, to block-encode the Pauli-Breit Hamiltonian, reducing T-gate count and improving resource utilization through double-factorization and quantum phase estimation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If current quantum circuit designs are used to simulate Pauli-Breit Hamiltonians, then the simulation can be performed, but the quantum T-gate complexity becomes excessively high
Solution Approach 1:
The Pauli-Breit Hamiltonian is decomposed into multiple Pauli terms that can be independently simulated. The Hamiltonian H = Σ c_i P_i is segmented into individual Pauli operators P_i, each of which can be simulated separately using quantum phase estimation, reducing the overall computational complexity compared to treating the entire Hamiltonian as a single unit.
Solution Approach 2:
The patent introduces an intermediary quantum circuit design that uses a linear combination of unitaries approach. Instead of directly simulating the complex Pauli-Breit Hamiltonian, the method uses intermediate quantum operations including qubitization and quantum phase estimation as mediators to achieve the simulation with reduced T-gate complexity.
2Measurement precision
If existing quantum circuit methods are used for eigenvalue estimation, then eigenvalues can be obtained, but resource utilization is inefficient
Solution Approach 1:
The patent applies preliminary action by preparing quantum states in advance using quantum phase estimation before performing the actual eigenvalue measurement. The quantum circuit pre-computes phase information during the QPE process, which is then used to extract eigenvalues, thereby improving measurement precision while optimizing resource usage compared to direct measurement methods.
Solution Approach 2:
The method changes the parameter representation by using a linear combination of unitaries where the Hamiltonian is expressed as H = Σ α_i U_i. This parameter transformation allows the eigenvalue problem to be solved through quantum phase estimation with optimized resource requirements, achieving better precision-to-resource ratio than traditional approaches.
Data Source
AI summary
A quantum circuit comprises: a first circuit primitive configured to prepare first qubits representing orbital indices in accordance with a first primitive matrix operation; a second circuit primitive configured to prepare second qubits representing spin indices in accordance with a second primitive matrix operation, wherein the spin indices are decoupled from the plurality of orbital indices; a plurality of spin-mixing swap networks configured to control the first qubits representing the orbital indices and the second qubits representing the spin indices; a third circuit primitive configured to implement, after the plurality of spin-mixing swap networks, a Hermitian conjugate of the first primitive matrix operation on the first qubits; and a fourth circuit primitive configured to implement, after the spin-mixing swap networks, a Hermitian conjugate of the second primitive matrix operation on the second qubits; wherein states of the first and second qubits estimate the eigenvalues of a PB Hamiltonian.