The entanglement-assisted Hadamard test (EHT) is a practical method for estimating a quantum phase by amplifying the phase signal. However, the feasible amplification is fundamentally limited by the accuracy of the reference phase, such that the method is inefficient in the high-precision regime. In this work, we propose an algorithm, called adaptive entanglement-assisted Hadamard test (AEHT), that iteratively refines the reference phase, enabling progressively stronger amplification as the iteration goes by. We further consider the imperfect eigenstate preparation scenario, where a systematic bias is unavoidable when estimating the quantum phase with the conventional EHT. Such a bias can be suppressed by the proposed AEHT. Moreover, taking physical implementation into consideration, we adopt device-restart count to measure the cost of quantum phase estimation, rather than shot count. The numerical experiments confirm the effectiveness of the proposed AEHT compared with conventional methods under this measure. By unlocking the full amplification power of large entangled states, this work offers an efficient method to estimate high-precision quantum phase on near-term quantum processors.
Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.
Quantum Phase Estimation (QPE) has potential for a scientific revolution through numerous practical applications like finding better medicines, batteries, materials, catalysts etc. Many QPE algorithms use the Hadamard test to estimate
⟨ψ∣Uk∣ψ⟩ for a large integer
k for an efficiently preparable initial state
∣ψ⟩ and an efficiently implementable unitary operator
U. The Hadamard test is hard to implement because it requires controlled applications of
Uk. Recently, a Sequential Hadamard test (SHT) was proposed (arXiv:2506.18765) which requires controlled application of
U only but its total run time
Ttot scales as
O(k3/ε2rmin2) where
rmin is the minimum value of
∣⟨ψ∣Uk′∣ψ⟩∣ among all integers
k′≤k. Typically
rmin is exponentially low and SHT becomes too slow. We present a new algorithm, the SANDWICH test to address this bottleneck. Our algorithm uses efficient preparation of the initial state
∣ψ⟩ to efficiently implement the SPROTIS operator
Rψϕ where SPROTIS stands for the Selective Phase Rotation of the Initial State. It sandwiches the SPROTIS operator between
Ua and
Ub for integers
{a,b}≤k to estimate
⟨ψ∣Uk∣ψ⟩. The total run time
Ttot is
O(k2lnk/ε2smin6). Here
smin is the minimum value of
∣⟨ψ∣Uk^∣ψ⟩ among all integers
k^ which are values of the nodes of a random binary sum tree whose root node value is
k and leaf nodes' values are
1 or
0. It can be reasonably expected that
smin≪1 in typical cases because there is wide freedom in choosing the random binary sum tree.
Estimating quantum time-series such as the Loschmidt amplitude
f(t)=⟨ψ∣e−iHt∣ψ⟩ is central to spectroscopy, Hamiltonian analysis, and many phase-estimation algorithms. Direct estimation via the Hadamard test requires controlled implementations of
e−iHt, and the depth of these controlled circuits grows with
t, making long-time estimation challenging on near-term hardware. We introduce Quantum Phaselift, a lifting-based framework that estimates the rank-one matrix
Z=ff† rather than estimating
f directly. We propose simple quantum circuits for estimating the entries of
Z and show that measuring only a narrow band of this matrix around the diagonal is sufficient to uniquely recover
f. Crucially, this reformulation decouples the controlled circuit depth from the maximum evolution time to scale instead with the width of the measured band. We prove that a
O(1) bandwidth suffices for generic signals, leading to substantial savings in controlled operations compared to direct estimation methods. We develop three recovery algorithms with provable exact recovery in the noiseless setting and stability under measurement noise. Finally, we numerically demonstrate that high-quality recovery is possible for the 2D Fermi-Hubbard and 2D transverse-field Ising model signals of size exceeding 100 time points using only a few million measurement shots and reasonable post-processing time, making our time-series estimation techniques efficient and effective for near-term implementations.
We present a comprehensive and pedagogical study of two core quantum algorithms in quantum linear algebra: the Hadamard test (HT) and quantum phase estimation (QPE).Building on the lecture notes on Quantum Algorithms for Scientific Computing [1] by Lin, we detail their mathematical formulations, circuit implementations, and convergence behavior.Both algorithms are benchmarked on simulators and IBM's 127-qubit Eagle processor (ibm rensselaer) using a simple eigenvalue problem.While QPE performs well in simulation, its hardware performance degrades beyond t = 3 ancilla qubits due to circuit depth and two-qubit gate error.In contrast, HT shows stable samplingbased convergence and greater resilience to hardware noise.These results provide a realistic baseline for implementing phase estimation on noisy intermediate-scale quantum (NISQ) era devices.
In this work we demonstrate the use of adapted classical phase retrieval algorithms to perform control-free quantum phase estimation. We eliminate the costly controlled time evolution and Hadamard test commonly required to access the complex time-series needed to reconstruct the spectrum. This significant reduction of the number of coherent controlled-operations lowers the circuit depth and considerably simplifies the implementation of statistical quantum phase estimation in near-term devices. This seemingly impossible task can be achieved by extending the problem that one wishes to solve to one with a larger set of input signals while exploiting natural constraints on the signal and/or the spectrum. We leverage well-established algorithms that are widely used in the context of classical signal processing, demonstrating two complementary methods to do this, vectorial phase retrieval and two-dimensional phase retrieval. We numerically investigate the feasibility of both approaches for estimating the spectrum of the Fermi-Hubbard model and discuss their resilience to inherent statistical noise.
We present a comprehensive and pedagogical study of two core quantum algorithms in quantum linear algebra: the Hadamard test (HT) and quantum phase estimation (QPE). Building on Prof. Lin's lecture notes on Quantum Algorithms for Scientific Computing, we detail their mathematical formulations, circuit implementations, and convergence behavior. Both algorithms are benchmarked on simulators and IBM's 127-qubit Eagle processor (ibm rensselaer) using a simple eigenvalue problem. While QPE performs well in simulation, its hardware performance degrades beyond t = 3 ancilla qubits due to circuit depth and two-qubit gate noise. In contrast, HT shows stable, sampling-based convergence and greater resilience to hardware noise. We further apply QPE to the transversefield Ising model (TFIM), demonstrating the importance of ansatz quality for accurate ground state energy estimation. These results provide a realistic baseline for implementing phase estimation on noisy intermediate-scale quantum (NISQ) era devices.