Quantum Phase Estimation (QPE) routines are known to fail probabilistically even with perfect gates and input states. This effect stems from an incompatibility of finite-sized quantum registers to capture a phase within QPE with phase angles of infinite precision, and the effect extend even beyond what would be reasonably expected from rounding. This effect can be partially mitigated by biasing the phase register with a window, or taper state, from classical signal processing. This paper focuses on how windowing a coherent QPE used as a subroutine can improve the accuracy of the overall algorithm. Specifically we study the quantum task of estimating observables where window-assisted coherent QPE is used as a subroutine to implement a reflection about an eigenstate. Quantum resource estimates show over 2-orders-of-magnitude reduction in Toffoli counts over the previous costed techniques -- also assisted by the use of improved block encoding techniques -- demonstrating an encouraging decrease in resources for quantum computation of molecular observables. Since QPE, as one of only a few quantum building blocks, appears as a subroutine in many algorithms; this analysis also provides a model for understanding how window functions propagate to an improved error in composite algorithms.
Quantum Phase Estimation (QPE) routines are known to fail probabilistically even with perfect gates and input states. This effect stems from an incompatibility of finite-sized quantum registers to capture a phase within QPE with phase angles of infinite precision, and the effect extend even beyond what would be reasonably expected from rounding. This effect can be partially mitigated by biasing the phase register with a window, or taper state, from classical signal processing. This paper focuses on how windowing a coherent QPE used as a subroutine can improve the accuracy of the overall algorithm. Specifically we study the quantum task of estimating observables where window-assisted coherent QPE is used as a subroutine to implement a reflection about an eigenstate. Quantum resource estimates show over 2-orders-of-magnitude reduction in Toffoli counts over the previous costed techniques -- also assisted by the use of improved block encoding techniques -- demonstrating an encouraging decrease in resources for quantum computation of molecular observables. Since QPE, as one of only a few quantum building blocks, appears as a subroutine in many algorithms; this analysis also provides a model for understanding how window functions propagate to an improved error in composite algorithms.
Quantum phase estimation (QPE) is a key ingredient to some of the most important results of quantum computing. In this paper we examine the QPE algorithm through the looking glass of signal processing. This perspective helps to reveal a relation between the distribution of quantum phase estimates and the shape of the discrete Fourier transform (DFT) of a pure complex sinusoid. Guided by spectral shaping theory, we design a class of quantum circuits that are the quantum equivalents to classical windowing operators, for a family of window functions that have compact support in the DFT. These circuits have complexity linear in the number of target qubits, strictly lower than QPE itself. Using these circuits we are able to reshape QPE's performance characteristics. In particular, the asymptotic marginal cost for 2x reduction in fail rate (2x improvement in reliability) is reduced from one qubit to 1/3 qubit or lower, which can help reduce the qubit budget for a preset accuracy-reliability target, especially in high-reliability scenarios. This makes a demonstration of the value of signal processing principles in designing quantum algorithms.
ABSTRACT Quantum Phase Estimation (QPE) is a cornerstone algorithm in quantum computing, with applications ranging from integer factorization to quantum chemistry simulations. However, the resource demands of standard QPE, which require a large number of coherent qubits and deep circuits, pose significant challenges for current Noisy Intermediate Scale Quantum (NISQ) devices. In this work, we introduce the Adaptive Windowed Quantum Phase Estimation (AWQPE) algorithm, a novel method designed to address the limitations of standard QPE. AWQPE utilizes small, independent blocks of control qubits to estimate multiple phase bits simultaneously within a “window,” thereby significantly reducing the number of coherent qubits required to achieve a desired precision. These independent blocks are amenable to parallelization and, when combined with a robust least‐significant‐bit (LSB) to most‐significant‐bit (MSB) ambiguity resolution mechanism, enhance the algorithm's accuracy while mitigating the risk of error propagation. Our numerical simulations demonstrate AWQPE's accuracy and improved robustness relative to standard QPE under representative noise models, showcasing a favorable balance between resource efficiency and computational speed. This makes AWQPE particularly well‐suited for near‐term quantum platforms.
Quantum phase estimation is one of the fundamental primitives that underpins many quantum algorithms, including Shor’s algorithm for efficiently factoring large numbers. Due to its significance as a subroutine, in this work, we consider the coherent version of the phase estimation problem, where given an arbitrary input state and black-box access to unitaries U and controlled- U , the goal is to estimate the phases of U in superposition. Most existing phase estimation algorithms involve intermediary measurements that disrupt coherence. Only a couple of algorithms, including the standard quantum phase estimation algorithm, consider this coherent setting. However, the standard algorithm only succeeds with a constant probability. To boost this success probability, one can employ the coherent median technique, resulting in an algorithm with asymptotically optimal query complexity (the total number of calls to U and controlled- U ). However, this coherent median technique requires a large number of ancilla qubits and a computationally expensive quantum sorting network. To address this, in this work, we propose an improved version of the standard algorithm called the tapered quantum phase estimation algorithm, which leverages tapering (or window) functions commonly used in classical signal processing. Our algorithm achieves the asymptotically optimal query complexity without requiring the expensive coherent median technique to boost success probability. Moreover, we find the absolutely optimal taper—not only in the asymptotic scaling but in terms of exact performance. We provide an efficiently preparable ancilla state based on an approximation of the optimal taper, which incurs at most a factor-of-two increase in the probability of error, thereby maintaining near-optimal performance in practice. In the Appendixes, we give an explicit construction of the taper state preparation circuit. Finally, we derive an error bound for coherent quantum phase estimation when the phase estimate is used as a control and subsequently uncomputed.
We provide a modification to the quantum phase estimation algorithm (QPEA) [Abrams and Lloyd, Phys. Rev. Lett. 83, 5162 (1999); Cleve et al., Proc. R. Soc. A 454, 339 (1998); Nielsen and Chuang, Quantum computation and quantum information, 2002.] inspired by classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm's error rate. Numerical evaluation of the costs also demonstrates an improvement. Moreover, with similar techniques, we detail an iterative projective measurement method for ground state preparation that gives an exponential improvement over previous bounds using QPEA. Numerical tests that confirm the expected scaling behavior are also obtained. For these numerical tests we have used a lattice Thirring model as testing ground. Using well-known perturbation theory results, we also show how to more appropriately estimate the cost scaling with respect to state error instead of evolution operator error.
High Resolution Image Download MS PowerPoint Slide Quantum phase estimation (QPE) is a promising quantum algorithm for obtaining molecular ground-state energies with chemical accuracy. However, its computational cost, dominated by the Hamiltonian 1-norm λ and the cost of the block encoding, scales at least quadratically with the number of molecular orbitals, making it challenging to incorporate dynamic correlation by enlarging the active space. In this work, we investigate two strategies to mitigate this cost through the optimization of the basis set. First, we investigate whether adjusting the coefficients of Gaussian basis functions can minimize the 1-norm while preserving the accuracy of the ground state energy. Although this method leads to a reduction in the 1-norm up to 10%, this reduction is system-dependent and diminishes with increasing molecular size. Second, we demonstrate that employing a large-basis-set frozen natural orbital (FNO) strategy results in a substantial reduction in QPE resources without compromising accuracy. We study a data set of 58 small organic molecules and the dissociation curve of N 2, and demonstrate that an active space constructed from orbitals derived from larger basis sets captures correlation effects more effectively. This approach yields up to an 80% reduction in the 1-norm λ and also leads to a 55% reduction in the number of orbitals. Our results highlight that improving the quality, not just the size, of the orbital basis is a viable strategy for extending QPE to include dynamical correlation, making progress toward scalable and chemically accurate quantum simulations with tractable resource requirements.
We present a hardware demonstration and resource analysis of split-evolution quantum phase estimation (SE-QPE) on a Quantinuum System Model H2 quantum computer. SE-QPE is a modification to canonical QPE for particle-conserving Hamiltonians in which controlled time evolution is replaced by CSWAP-based interference between a target register and a reference register. For factorizations of time evolution with a shared eigenbasis, SE-QPE preserves the phase-register outcome distribution of canonical QPE and, unlike with compute--uncompute substitutions, it remains compatible with non-exact eigenstates. The substitution removes controlled-simulation overhead and enables parallel evolution on two registers, reducing the depth of each phase-kickback block. Resource analysis for Trotterized double-factorized chemistry Hamiltonians shows that the substitution becomes increasingly favorable at higher phase powers and combining QPE and SE-QPE implementations can be a useful option. Over a range of FeMoco active spaces, SE-QPE reduces time evolution resources, with asymptotic reductions of about 33% in CX count, 25% in T count, and an asymptotic depth ratio of 3/N for CX layers. On Quantinuum H2-2, a four-qubit model ethylene demonstration with explicit inverse QFT and repeated phase-kickback steps up to 8 phase bits yields distinct energies and shows the auxiliary registers provide useful error detection filters.