We develop an energy calculation algorithm leveraging quantum phase difference estimation (QPDE) scheme and a tensor-network-based unitary compression method in the preparation of superposition states and time-evolution gates. Alongside its efficient implementation, this algorithm reduces depolarization noise affections exponentially. We demonstrated energy gap calculations for one-dimensional Hubbard models on IBM superconducting devices using circuits up to 32-system (plus one-ancilla) qubits, a five-fold increase over previous Quantum phase estimation (QPE) demonstrations, at the 7242 controlled-Z gate level of standard transpilation, utilizing a Q-CTRL error suppression module. Additionally, we propose a technique toward molecular executions using spatial orbital localization and index sorting, verified linear polyene simulations up to 21 qubits. Since QPDE can handle the same objectives as QPE, our algorithm represents a leap forward in quantum computing on real devices.
We develop an energy calculation algorithm leveraging quantum phase difference estimation (QPDE) scheme and a tensor-network-based unitary compression method in the preparation of superposition states and time-evolution gates. Alongside its efficient implementation, this algorithm reduces depolarization noise affections exponentially. We demonstrated energy gap calculations for one-dimensional Hubbard models on IBM superconducting devices using circuits up to 32-system (plus one-ancilla) qubits, a five-fold increase over previous Quantum phase estimation (QPE) demonstrations, at the 7242 controlled-Z gate level of standard transpilation, utilizing a Q-CTRL error suppression module. Additionally, we propose a technique toward molecular executions using spatial orbital localization and index sorting, verified linear polyene simulations up to 21 qubits. Since QPDE can handle the same objectives as QPE, our algorithm represents a leap forward in quantum computing on real devices.
We propose a phase-difference estimation algorithm based on the tensor-network circuit compression, leveraging time-evolution data to pursue scalability and higher accuracy on a quantum phase estimation (QPE)-type algorithm. Using tensor networks, we construct circuits composed solely of nearest-neighbor gates and extract time-evolution data by four-type circuit measurements. In addition, to enhance the accuracy of time-evolution and state-preparation circuits, we propose techniques based on algorithmic error mitigation and on iterative circuit optimization combined with merging into matrix product states, respectively. Verifications using a noiseless simulator for the 8-qubit one-dimensional Hubbard model using an ancilla qubit show that the proposed algorithm achieves accuracies with 0.4--4.7\% error from a true energy gap on an appropriate time-step size, and that accuracy improvements due to the algorithmic error mitigation are observed. We also confirm the enhancement of the overlap with matrix product states through iterative optimization. Finally, the proposed algorithm is demonstrated on IBM Heron devices with Q-CTRL error suppression for 8-, 36-, and 52-qubit models using more than 4,000 2-qubit gates. These largest-scale demonstrations for the QPE-type algorithm represent significant progress not only toward practical applications of near-term quantum computing but also toward preparation for the era of error-corrected quantum devices.
The Quantum Phase Difference Estimation (QPDE) algorithm, as an extension of the Quantum Phase Estimation (QPE), is a quantum algorithm designed to compute the differences of two eigenvalues of a unitary operator by exploiting the quantum superposition of two eigenstates. Unlike QPE, QPDE is free of controlled-unitary operations, and is suitable for calculations on noisy intermediate-scale quantum (NISQ) devices. We present the implementation and verification of a novel early fault-tolerant QPDE algorithm for determining energy gaps across diverse spin system configurations using NISQ devices. The algorithm is applied to the systems described by two and three-spin Heisenberg Hamiltonians with different geometric arrangements and coupling strengths, including symmetric, asymmetric, spin-frustrated, and non-frustrated configurations. By leveraging the match gate-like structure of the time evolution operator of Heisenberg Hamiltonian, we achieve constant-depth quantum circuits suitable for NISQ hardware implementation. Our results on IBM quantum processors show remarkable accuracy ranging from 85\% to 93\%, demonstrating excellent agreement with classical calculations even in the presence of hardware noise. The methodology incorporates sophisticated quantum noise suppression techniques, including Pauli Twirling and Dynamical Decoupling, and employs an adaptive framework. Our findings demonstrate the practical viability of the QPDE algorithm for quantum many-body simulations on current NISQ hardware, establishing a robust framework for future applications.
Accurate calculation of spin-state energy gaps is central to spin chemistry. The novel Quantum Phase Difference Estimation (QPDE) algorithm enables direct computation of energy gaps on a quantum computer. However, the required quantum circuits are typically too deep for noisy intermediate-scale quantum (NISQ) devices. In this study, as an initial step toward practical calculations for strongly correlated molecular multi-spin systems, we applied QPDE to two- and three-spin Heisenberg Hamiltonians with various geometries and coupling strengths, including symmetric, asymmetric, spin-frustrated, and non-frustrated configurations. We found that the quantum circuit for the time-evolution operator achieves constant depth due to its match gate-like structure, making it well-suited for NISQ implementation. Proof-of-principle hardware demonstrations using an IBM quantum processor yielded 85%-96% accuracy in determining spin-state energy gaps.
Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.
Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision ε in Heisenberg-limited time T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling T=O(ε1log(δ−1)) in both the precision ε and failure probability δ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.