Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices
Under suitable assumptions, some recently developed quantum algorithms can estimate the ground-state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block-encoding input model of the Hamiltonian, the implementation of whi...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2022-10-01
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| Series: | PRX Quantum |
| Online Access: | http://doi.org/10.1103/PRXQuantum.3.040305 |
| _version_ | 1817985744914874368 |
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| author | Yulong Dong Lin Lin Yu Tong |
| author_facet | Yulong Dong Lin Lin Yu Tong |
| author_sort | Yulong Dong |
| collection | DOAJ |
| description | Under suitable assumptions, some recently developed quantum algorithms can estimate the ground-state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block-encoding input model of the Hamiltonian, the implementation of which is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QETU), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit, and no multiqubit control operations and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground-state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anticommutation relations and further removes the need to implement the controlled Hamiltonian evolution. Coupled with a Trotter-based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM qiskit for the transverse-field Ising model. If we are further allowed to use multiqubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude-estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground-state preparation and energy estimation using a constant number of ancilla qubits (no more than three). |
| first_indexed | 2024-04-14T00:01:01Z |
| format | Article |
| id | doaj.art-304e8f5e64f949dc94d6c79908cc1312 |
| institution | Directory Open Access Journal |
| issn | 2691-3399 |
| language | English |
| last_indexed | 2024-04-14T00:01:01Z |
| publishDate | 2022-10-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | PRX Quantum |
| spelling | doaj.art-304e8f5e64f949dc94d6c79908cc13122022-12-22T02:23:41ZengAmerican Physical SocietyPRX Quantum2691-33992022-10-013404030510.1103/PRXQuantum.3.040305Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary MatricesYulong DongLin LinYu TongUnder suitable assumptions, some recently developed quantum algorithms can estimate the ground-state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block-encoding input model of the Hamiltonian, the implementation of which is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QETU), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit, and no multiqubit control operations and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground-state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anticommutation relations and further removes the need to implement the controlled Hamiltonian evolution. Coupled with a Trotter-based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM qiskit for the transverse-field Ising model. If we are further allowed to use multiqubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude-estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground-state preparation and energy estimation using a constant number of ancilla qubits (no more than three).http://doi.org/10.1103/PRXQuantum.3.040305 |
| spellingShingle | Yulong Dong Lin Lin Yu Tong Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices PRX Quantum |
| title | Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices |
| title_full | Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices |
| title_fullStr | Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices |
| title_full_unstemmed | Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices |
| title_short | Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices |
| title_sort | ground state preparation and energy estimation on early fault tolerant quantum computers via quantum eigenvalue transformation of unitary matrices |
| url | http://doi.org/10.1103/PRXQuantum.3.040305 |
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