[Paper Review] Ground state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices

Yulong Dong, Lin Lin+1 more|arXiv (Cornell University)|Apr 12, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a new quantum algorithm, Quantum Eigenvalue Transformation of Unitary matrices (QET-U), that enables ground-state preparation and energy estimation on early fault-tolerant quantum computers using only a single ancilla qubit and controlled Hamiltonian evolution, avoiding costly multi-qubit controls. The method achieves near-optimal query complexity with reduced circuit depth, outperforming prior approaches under realistic hardware constraints.

ABSTRACT

Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] 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, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit 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 anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with 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 multi-qubit 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 estimating using a constant number of ancilla qubits (no more than 3).

Motivation & Objective

• Address the high resource overhead of block encoding-based methods for ground-state preparation and energy estimation on early fault-tolerant quantum computers.
• Develop a practical input model based on controlled Hamiltonian evolution to reduce circuit depth and ancilla qubit requirements.
• Enable efficient ground-state energy estimation and state preparation using only one ancilla qubit and no multi-qubit control gates.
• Optimize the algorithm for sparse quantum spin Hamiltonians by exploiting anti-commutation relations to eliminate controlled Hamiltonian evolution.
• Demonstrate the feasibility and performance of the algorithm using Qiskit simulations on the transverse field Ising model.

Proposed method

• Propose the Quantum Eigenvalue Transformation of Unitary matrices (QET-U) using real polynomials to transform the eigenvalues of a unitary matrix derived from Hamiltonian evolution.
• Use a single ancilla qubit and controlled time evolution $ U = e^{-i\tau H} $ as input, avoiding the need for block encoding and multi-qubit control operations.
• Implement the algorithm via a controlled rotation protocol that encodes the energy information in the measurement probability of the ancilla qubit.
• Leverage Trotter-based approximation of the Hamiltonian evolution to make the method suitable for near-term fault-tolerant devices.
• Integrate amplitude amplification and a new binary amplitude estimation algorithm when multi-qubit Toffoli gates are available, reducing total query complexity.
• Use marginal probability estimation from measurements to compute the ground-state energy via signed linear combinations of Pauli expectation values.

Experimental results

Research questions

• RQ1 Can ground-state energy estimation and preparation be achieved with near-optimal query complexity while minimizing circuit depth and ancilla qubit usage on early fault-tolerant devices?
• RQ2 Can the controlled Hamiltonian evolution model replace block encoding to reduce resource overhead in quantum algorithms?
• RQ3 Can anti-commutation relations in specific Hamiltonians (e.g., transverse field Ising model) be exploited to eliminate the need for controlled evolution?
• RQ4 How does the performance of the QET-U algorithm compare to prior methods in terms of query complexity and circuit depth for realistic quantum hardware constraints?
• RQ5 Can the energy estimation be efficiently computed from measurement data using marginal probabilities without full state tomography?

Key findings

• The QET-U algorithm achieves near-optimal query complexity for ground-state energy estimation and preparation using only one ancilla qubit and no multi-qubit control gates.
• For the transverse field Ising model, the algorithm enables energy estimation by measuring marginal probabilities from two circuits: one for the ground state and one after applying Hadamard gates to all qubits.
• The method reduces circuit depth significantly compared to block encoding-based approaches, making it suitable for early fault-tolerant devices with limited coherence time.
• By exploiting anti-commutation relations in certain Hamiltonians, the need for controlled Hamiltonian evolution is eliminated, further simplifying the circuit.
• When Toffoli gates are available, amplitude amplification and a new binary amplitude estimation algorithm reduce total query complexity while increasing circuit depth.
• Numerical results using Qiskit show that the algorithm performs well on the transverse field Ising model, with ground-state energy estimated accurately from measurement frequencies and marginal probabilities.