Purified phase estimation samples spectra efficiently

TL;DR

This work addresses the eigenstate-preparation bottleneck of quantum phase estimation by introducing DOS-QPE, a purification-based variant that uses mixed-state probes to sample the density of states of a Hamiltonian. By incorporating a purification register and simple entangling layers, the method converts standard QPE into an ensemble-based DOS sampler, enabling spectrum reconstruction through convex optimization and compressed sensing. The authors formalize the purified DOS-QPE framework, develop symmetry-adapted input ensembles (maximally mixed and Dicke-state probes), and demonstrate spectral recovery in the Fermi-Hubbard model, electronic structure, and nuclear-structure problems, even with limited time-frequency resolution. The approach offers a practical pathway to extract thermodynamic and spectral information from ensembles on near-term quantum devices, with potential extensions to Green's-function formulations and broader quantum-learning paradigms.

Abstract

Quantum phase estimation (QPE) is a cornerstone algorithm for extracting Hamiltonian eigenvalues, but its standard, eigenstate-centric form relies on carefully prepared coherent inputs that are costly or impractical for many strongly correlated systems. We overcome this bottleneck via DOS-QPE, an incoherent, purification-based variant of QPE that works directly with mixed-state probes and estimates the density of states (DOS) of the Hamiltonian. By adding a purification register and simple entangling layers, we turn standard QPE into an ensemble-based DOS sampler without modifying the core phase-estimation block. Conceptually, this purification closely aligns with the recent random purification channel framework from quantum learning theory, but instantiated here as a concrete phase-estimation circuit. We further equip DOS-QPE with symmetry-adapted input ensembles and a compressed-sensing reconstruction pipeline, and demonstrate on fermionic and nuclear Hamiltonians that a single experimental setup can recover rich spectral information relevant to thermodynamics, spectroscopy, and many-body structure.

Figures (5)

• Figure 1: a) DOS-QPE circuit primitive. In panel a), we show the DOS-QPE as a circuit primitive. The full circuit consists of three registers: a time-frequency register, a state register, and a purification register. The additional purification register is maximally entangled to the state register to create the state ensemble. This is done through a cascade of serialized CNOT operations, $CX^{\otimes n}_{S\rightarrow P}$. This last register is traced out at the end of the protocol. b) QPE v. DOS-QPE algorithmic comparison. In panel b), we compare the standard QPE algorithm to the DOS-QPE algorithm. While QPE relies on the preparation of good-enough states $|\psi\rangle$ with sufficient overlap with the target eigenstates, DOS-QPE allows for probing entire subspaces via purifications $|\psi_{SP}\rangle$ of simple construction. This translates to a cheaper state preparation routine. After measuring the time-frequency register in QPE, one estimates the phase $\theta$. In DOS-QPE, we measure the time-frequency register and trace out the purification register to estimate the eigenvalue function $S(\theta)$.
• Figure 2: DOS-QPE on the Fermi-Hubbard model. We run the DOS-QPE circuit on a Fermi-Hubbard toy model with three sites, $t=1$, and $U=4$, employing $m=6$ ancillas to resolve the spectrum. In the time-frequency register, we obtain the probability distribution of the DOS (blue bars). We compare this with the exact spectrum (black bars) of the Hamiltonian. While the dimensionality of the time-frequency register is insufficient to resolve the spectral gaps of the spectrum, we show that quadratic programming aided by compressed sensing helps in the phase and degeneracy reconstruction (red stars). The 1-Wasserstein distance of the optimized solution improved by $\sim51\%$ over the integer-rounded sampled distribution.
• Figure 3: DOS-QPE on the electronic structure of ethylene. a) The structure of the ethylene molecule showing a twisted conformation and the angle $\alpha$ which is varied during the scan of the potential energy surfaces. b) Plot of the lowest three eigenvalues of the Hamiltonian, scaled such that the eigenvalues lie in the interval $[0, 1)$. The green line corresponds to the triplet state $T$, the orange line to the lowest lying singlet state S$_0$ and the purple line to the first excited singlet state S$_1$. c--f) The distributions obtained with DOS-QPE with $\alpha= 65^\circ, 68^\circ, 71^\circ$ and $74^\circ$, respectively. In each plot c--f), the colour of each vertical dashed line corresponds to the phases obtained via exact diagonalization and matches those used in b). The height of the dashed lines shows the height the peaks in the distribution would be if all eigenvalues were to be exact integer multiples of $\frac{1}{N_\text{anc}}$.
• Figure 4: a) Many-body state basis used for ${}^6$Be. Label represents the indices in the corresponding Hamiltonian matrix in this many-body basis. The many-body states are composed of single-particle states from all possible combinations of two valence protons in the $p-$shell variational space of panel b) satisfying the condition $M=m_1+m_2=0$. b) Configurational valence space for the $p-$shell. The valence space is decomposed in two degenerate $j$ orbitals with degeneracy $(2j+1)$. The labels on the left correspond to the standard spectroscopic notation $n\ell_j$, where $\ell=p$ means $\ell=1$. Numbers on top of every single-particle state represent the labels used in this work. Slightly modified from perez2023nuclear.
• Figure 5: DOS-QPE on $^6$Be. The result of DOS-QPE with a maximally mixed state probe. Sixth-order Trotterization of the unitary evolution and five qubits in the time-frequency register were used for these results. We observe that, while the distribution (blue bars) faithfully captures the exact eigenspectrum, it is not fine-grained enough to resolve the almost-degenerate peaks at phase $\sim 0.65$, and more ancillary qubits would be needed. The tallest peak represents the redundant three-times degenerate zero eigenvalue resulting from building the nuclear Hamiltonian matrix using the LCU approach.