[Paper Review] Quantum algorithm for finding periodicities in the spectrum of a black-box Hamiltonian or unitary transformation

Dominik Janzing, Thomas Beth|ArXiv.org|Aug 10, 2001

Quantum Computing Algorithms and Architecture6 references3 citations

TL;DR

This paper presents a quantum algorithm to estimate periodicities in the energy spectrum of an unknown Hamiltonian by measuring eigenvalues of $U \otimes U^\dagger$ using only black-box access to the unitary $U = \exp(-iHt)$. By leveraging state exchange and controlled operations on auxiliary registers, the method bypasses the need for controlled-$U$ gates, enabling phase estimation on $U \otimes U^\dagger$ and thus extracting the autocorrelation function of the density of states, which reveals spectral periodicities.

ABSTRACT

Estimating the eigenvalues of a unitary transformation U by standard phase estimation requires the implementation of controlled-U-gates which are not available if U is only given as a black box. We show that a simple trick allows to measure eigenvalues of U\otimes U^\dagger even in this case. Running the algorithm several times allows therefore to estimate the autocorrelation function of the density of eigenstates of U. This can be applied to find periodicities in the energy spectrum of a quantum system with unknown Hamiltonian if it can be coupled to a quantum computer.

Motivation & Objective

• To address the limitation of standard quantum phase estimation, which requires controlled-$U$ operations not available when $U$ is a black box.
• To enable spectral analysis of unknown Hamiltonians in quantum systems, such as many-body systems in solid-state physics.
• To extract the autocorrelation function of the density of eigenstates via eigenvalue estimation of $U \otimes U^\dagger$ without requiring explicit knowledge of $U$'s structure.
• To provide a practical method for detecting periodicities in energy spectra—such as regular spectral gaps—using only unitary evolution and ancilla-based control.

Proposed method

• Use a target register $\mathcal{H}$, an ancilla register $\mathcal{R}_a$ of $k$ qubits, and two auxiliary registers $\mathcal{R}_1$ and $\mathcal{R}_2$ of equal dimension to $\mathcal{H}$.
• Implement a conjugation protocol that transforms $U$ into an effective controlled-$U^{2^j}$ operation on $\mathcal{R}_1$ via state exchange and conditional operations.
• Apply the swap operation between $\mathcal{R}_1$ and $\mathcal{R}_2$ conditioned on the control qubit $j$ in $\mathcal{R}_a$, using a Fredkin gate for qubit-level implementation.
• Use the resulting transformation $V_j'' = |1_j\rangle\langle 1_j| \otimes U^{2^j} \otimes U^{-2^j} + |0_j\rangle\langle 0_j| \otimes 1 \otimes 1$ to perform phase estimation on $U \otimes U^\dagger$.
• Initialize the ancilla in a superposition state and apply the inverse quantum Fourier transform to extract eigenvalue information from the resulting state.
• Repeat the procedure multiple times to estimate the autocorrelation function of the density of eigenstates of $U$, which reflects periodic structures in the spectrum.

Experimental results

Research questions

• RQ1 Can periodicities in the energy spectrum of an unknown Hamiltonian be detected without access to controlled-$U$ operations?
• RQ2 Is it possible to estimate the autocorrelation function of the density of states using only black-box access to $U = \exp(-iHt)$?
• RQ3 How can phase estimation be adapted to work on $U \otimes U^\dagger$ instead of $U$ when controlled-$U$ gates are unavailable?
• RQ4 What physical information—such as spectral gaps or periodic energy level spacing—can be extracted from the eigenvalues of $U \otimes U^\dagger$?
• RQ5 Can this method be applied to many-body quantum systems where full spectral analysis is intractable?

Key findings

• The algorithm enables phase estimation on $U \otimes U^\dagger$ without requiring controlled-$U$ operations, overcoming a key limitation of standard phase estimation.
• By using state exchange and conditional swaps between auxiliary registers, the method effectively implements a controlled-$U^{2^j}$ operation on $\mathcal{R}_1$ via conjugation.
• The eigenvalues of $U \otimes U^\dagger$ correspond to the phase differences between eigenvalues of $U$, allowing the reconstruction of the autocorrelation function of the density of states.
• The method allows detection of periodicities in the energy spectrum, such as regular spectral gaps, even when the full spectrum is unknown or exponentially large.
• The required ancilla register size does not necessarily scale with system size if only periodic features (e.g., constant gap spacing) are of interest, enabling efficient detection in many-body systems.
• The approach is applicable to physical systems like solid-state materials where spectral periodicities influence thermodynamic and dynamical behavior.