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arXiv:1903.11999 · Physical Review A

Quantum algorithm for spectral projection by measuring an ancilla iteratively

Yanzhu Chen, Tzu-Chieh Wei

Published Mar 23, 2020• 7 citations
arXivHTMLPDF

Abstract

We propose a quantum algorithm for projecting a quantum system to eigenstates of any Hermitian operator, provided one can access the associated control-unitary evolution for the ancilla and the system, as well as the measurement of the controlling ancillary qubit. Such a Hadamard-test-like primitive is iterated so as to achieve the spectral projection, and the distribution of the projected eigenstates obeys the Born rule. This algorithm can be used as a subroutine in the quantum annealing procedure by measurement to drive the system to the ground state of a final Hamiltonian, and we simulate this for quantum many-body spin chains.

DOI
10.1103/physreva.101.032339

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We propose a quantum algorithm for projecting a quantum system to eigenstates of any Hermitian operator, provided one can access the associated control-unitary evolution for the ancilla and the system, as well as the measurement of the controlling ancillary qubit. Such a Hadamard-test-like primitive is iterated so as to achieve the spectral projection, and the distribution of the projected eigenstates obeys the Born rule. This algorithm can be used as a subroutine in the quantum annealing procedure by measurement to drive the system to the ground state of a final Hamiltonian, and we simulate this for quantum many-body spin chains.
Preparing the ground state of a system is an important task in physics. We propose a quantum algorithm for preparing the ground state of a physical system that can be simulated on a quantum computer. The system is coupled to an ancillary qubit, by introducing a resonance mechanism between the ancilla qubit and the system, and combined with measurements performed on the ancilla qubit, the system can be evolved to monotonically converge to its ground state through an iterative procedure. We have simulated the application of this algorithm for the Afflect-Kennedy-Lieb-Tasaki model, whose ground state can be used as resource state in one-way quantum computation.
Abstract We propose an algorithm to create an arbitrary n -qubit pure quantum superposition state with precision of m -decimals (binary representation) for each probability amplitude. The algorithm uses one-qubit rotations, Hadamard transformations and, C-NOT operations with multi-qubit controls. The depth of the circuit is O (2 n n ), its space is O ( n ). We emphasize that the parameters of the utilized unitary transformations are predicted in advance by the required precision and therefore there is no classical calculation supplementing this quantum algorithm. Finalizing the state-creation we perform the measurement of the ancilla state with a certain desired output with the purpose of removing all the garbage from the created superposition state. If n and m are independent and n ≫ m , the probability of access to the desired ancilla-state (success probability) is ~ 2 −n , which requires O (2 n ) runs of the algorithm and leads to the overall depth O (4 n n ). However, the situation can be significantly improved by replacing the usual measurement with the controlled measurement of the ancilla state that, first of all, removes the garbage part of the superposition state and, second (and most important), allows to avoid both the problem of small success probability to the desired ancilla state and multiple runs of the algorithm. As a consequence, the depth of the algorithm is O (2 n n ) in this case. If m serves to provide the required fidelity of state approximation, then the above parameters increase. This algorithm can be a subroutine generating the required input state in various algorithms, in particular, in matrix-manipulation algorithms developed earlier.
Quantum phase estimation (QPE) is a crucial subroutine in many quantum algorithms. It is used to estimate the eigenvalues of a unitary operator acting on a target quantum system. Conventional QPE is often expensive to implement because it requires a large number of ancilla qubits and the ability to perform quantum Fourier transform. Recent developments in iterative QPE reduce the implementation cost by repetitive uses of a single ancilla and classical postprocessing. However, both conventional and iterative schemes often require preparation of the target system in an eigenstate of the unitary operator, while it remains ambiguous to achieve QPE of multiple eigenvalues with no need of initial-state preparation. Here we clarify this issue by developing a theoretical framework based on sequential quantum channels for iterative QPE. We find that QPE of multiple eigenvalues can be efficiently realized for arbitrary initial target system state by actively utilizing the measurement backaction of iterative QPE on the target system with a long coherence time. Specifically, we investigate two iterative QPE schemes based on sequential Ramsey interferometry measurements (RIMs) of an ancilla qubit: (a) the repetitive scheme, which conducts repetitive RIMs to achieve the standard quantum limit in estimating the eigenvalues; (b) the adaptive scheme, which adjusts the parameters of each RIM based on prior measurement outcomes to attain the Heisenberg limit. In both schemes, sequential ancilla measurements generate sequential quantum channels on the target system, gradually steering it to the eigenstates of the estimated unitary operator, while the measurement statistics of the ancilla can reveal the embedded information about its eigenvalues with proper postprocessing. We demonstrate the analysis by simulating a central spin model and evaluate the performance and noise resilience of both schemes.
For finite-dimensional quantum systems, we propose a quantum control scheme based on a multi-step unitary evolution and quantum projective measurements. The objective is to design a control law to steer the system to a target eigenstate of the measurement operator in the least number of steps. Within each control step, unitary evolution and quantum projective measurement are performed in turn until the system reaches the target state. The control process can be modeled as a finite-state Markov chain with an absorbing state. We prove that the controlled system will converge to the target eigenstate with probability one after a finite number of control steps and find a minimal-step-number condition that would steer the system to the target eigenstate in the least number of steps.
Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop an algorithmic toolbox, quantum phase processing, that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigeninformation of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase related. As a notable application, we propose a quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropy estimation, demonstrating improvements or optimality for almost all cases.
We propose a quantum algorithm to obtain the lowest eigenstate of any Hamiltonian simulated by a quantum computer. The proposed algorithm begins with an arbitrary initial state of the simulated system. A finite series of transforms is iteratively applied to the initial state assisted with an ancillary qubit. The fraction of the lowest eigenstate in the initial state is then amplified up to ≃1\simeq 1≃1. We prove that our algorithm can faithfully work for any arbitrary Hamiltonian in the theoretical analysis. Numerical analyses are also carried out. We firstly provide a numerical proof-of-principle demonstration with a simple Hamiltonian in order to compare our scheme with the so-called Demon-like algorithmic (DLAC), recently proposed in [Nature Photonics 8, 113 (2014)]. The result shows a good agreement with our theoretical analysis, exhibiting the comparable behavior to the best cooling with the DLAC method. We then consider a random Hamiltonian model for further analysis of our algorithm. By numerical simulations, we show that the total number ncn_cnc​ of iterations is proportional to ≃O(D−1ϵ−0.19)\simeq {\cal O}(D^{-1}\epsilon^{-0.19})≃O(D−1ϵ−0.19), where DDD is the difference between the two lowest eigenvalues, and ϵ\epsilonϵ is an error defined as the probability that the finally obtained system state is in an unexpected (i.e. not the lowest) eigenstate.