Efficient Quantum Algorithm for Filtering Product States
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 Munich, Germany
| Published: | 2024-06-27, volume 8, page 1389 |
| Eprint: | arXiv:2312.13892v3 |
| Doi: | https://doi.org/10.22331/q-2024-06-27-1389 |
| Citation: | Quantum 8, 1389 (2024). |
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Abstract
We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width $\delta$. Given a local Hamiltonian on $N$ qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to $\delta\sqrt{N}$. We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in $\mathrm{poly}(N,1/\delta)$ time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width $\delta$ is independent of the system size $N$. Furthermore, the adiabatic evolution can be implemented with circuit depth $\mathcal{O}(N^2\delta^{-4})$. Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.
Featured image: Summary of our work is presented here. The left pane shows the application of the filter in the energy basis. The right pane shows how the filtering can be realzied using adiabatic evolution
Popular summary
In order to study the system at a particular energy one needs have direct access to the respective eigenstate. Obtaining such eigenstates is hard, however, the same physics can be extracted by studying a superposition of eigenstates that are close together, or in other words, have a small energy variance.
In this work, we present a quantum algorithm that allows to prepare states with a small energy variance. We achieve this by starting with a product state at the desired energy. These product states typically do not have a sufficiently small energy variance to explore finite energy properties. Our algorithm allows to suppress the energy eigenstates that are far away from the desired investigation energy, thus, reducing the energy variance. We call this process filtering.
The filtering is achieved by constructing an adiabatic evolution path that connects the initial product state to the filtered state. Thus, to prepare the filtered state on a quantum computer, one needs to perform a suitable time evolution that we construct. Additionally, we also prove that this evolution can be implemented with a quantum circuit depth that scales polynomially with the system size and the energy suppression parameter.
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[2] Xiaoyu Liu, Benjamin F. Schiffer, and Jordi Tura, "Preparing low-variance states using a distributed quantum algorithm", Quantum 9, 1838 (2025).
[3] Ke Liao, "Energy-Filtered Excited States and Real-Time Dynamics Served in a Contour Integral", Journal of Chemical Theory and Computation 21 12, 6074 (2025).
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[5] Erenay Karacan, Yanbin Chen, and Christian B. Mendl, "Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering", Quantum 9, 1624 (2025).
[6] Erenay Karacan, Conor Mc Keever, Michael Foss-Feig, David Hayes, and Michael Lubasch, "Filter-enhanced adiabatic quantum computing on a digital quantum processor", Physical Review Research 7 3, 033153 (2025).
[7] Alexander Schuckert, Or Katz, Lei Feng, Eleanor Crane, Arinjoy De, Mohammad Hafezi, Alexey V. Gorshkov, and Christopher Monroe, "Observation of a finite-energy phase transition in a one-dimensional quantum simulator", Nature Physics 21 3, 374 (2025).
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[9] Rei Sakuma, Kaito Wada, Shu Kanno, Kimberlee Keithley, Kenji Sugisaki, Takashi Abe, Hajime Nakamura, and Naoki Yamamoto, "Quantum-phase-estimation-based filtering: Performance analysis and application to low-energy spectral calculations", Physical Review A 113 1, 012602 (2026).
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[11] Ke Liao, "Energy-filtered excited states and real-time dynamics served in a contour integral", arXiv:2409.07354, (2024).
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