The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.
The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.
We present the problem of approximating the time-evolution operator e−iH^t to error ϵ , where the Hamiltonian H^=(⟨G|⊗I^)U^(|G⟩⊗I^) is the projection of a unitary oracle U^ onto the state |G⟩ created by another unitary oracle. Our algorithm solves this with a query complexity O(t+log(1/ϵ)) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d -sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H^ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H^ in an invariant SU(2) subspace. A large class of operator functions of H^ can then be computed with optimal query complexity, of which e−iH^t is a special case.
Simulating the unitary dynamics of a quantum system is a fundamental problem of quantum mechanics, in which quantum computers are believed to have significant advantage over their classical counterparts. One prominent such instance is the simulation of electronic dynamics, which plays an essential role in chemical reactions, non-equilibrium dynamics, and material design. These systems are time-dependent, which requires that the corresponding simulation algorithm can be successfully concatenated with itself over different time intervals to reproduce the overall coherent quantum dynamics of the system. In this paper, we quantify such simulation algorithms by the property of being fully-coherent: the algorithm succeeds with arbitrarily high success probability 1 − δ while only requiring a single copy of the initial state. We subsequently develop fully-coherent simulation algorithms based on quantum signal processing (QSP), including a novel algorithm that circumvents the use of amplitude amplification while also achieving a query complexity additive in time t, ln(1/δ), and ln(1/ϵ) for error tolerance ϵ: Θ‖H‖|t|+ln(1/ϵ)+ln(1/δ). Furthermore, we numerically analyze these algorithms by applying them to the simulation of the spin dynamics of the Heisenberg model and the correlated electronic dynamics of an H2 molecule. Since any electronic Hamiltonian can be mapped to a spin Hamiltonian, our algorithm can efficiently simulate time-dependent ab initio electronic dynamics in the circuit model of quantum computation. Accordingly, it is also our hope that the present work serves as a bridge between QSP-based quantum algorithms and chemical dynamics, stimulating a cross-fertilization between these exciting fields.
Hamiltonian simulation is a fundamental problem at the heart of quantum computation, and the associated simulation algorithms are useful building blocks for designing larger quantum algorithms. In order to be successfully concatenated into a larger quantum algorithm, a Hamiltonian simulation algorithm must succeed with arbitrarily high success probability 1−δ while only requiring a single copy of the initial state, a property which we call fully-coherent. Although optimal Hamiltonian simulation has been achieved by quantum signal processing (QSP), with query complexity linear in time t and logarithmic in inverse error ln(1/ϵ), the corresponding algorithm is not fully-coherent as it only succeeds with probability close to 1/4. While this simulation algorithm can be made fully-coherent by employing amplitude amplification at the expense of appending a ln(1/δ) multiplicative factor to the query complexity, here we develop a new fully-coherent Hamiltonian simulation algorithm that achieves a query complexity additive in ln(1/δ): Θ(∥H∥∣t∣+ln(1/ϵ)+ln(1/δ)). We accomplish this by compressing the spectrum of the Hamiltonian with an affine transformation, and applying to it a QSP polynomial that approximates the complex exponential only over the range of the compressed spectrum. We further numerically analyze the complexity of this algorithm and demonstrate its application to the simulation of the Heisenberg model in constant and time-dependent external magnetic fields. We believe that this efficient fully-coherent Hamiltonian simulation algorithm can serve as a useful subroutine in quantum algorithms where maintaining coherence is paramount.
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that |P|≤1 , a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 107 . We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ -approximate fractional query problem that requires O((1/δ)+log(1/ϵ)) queries to perform where O(1/δ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O(dlogN+log2N) 1 and 2-qubit gates for a filter of lengths d . Published by the American Physical Society 2024
Abstract Quantum signal processing (QSP) is a powerful toolbox for the design of quantum algorithms and can lead to asymptotically optimal computational costs. Its realization on noisy quantum computers without fault tolerance, however, is challenging because it requires a deep quantum circuit in general. We propose a strategy to run an entire QSP protocol on noisy quantum hardware by carefully reducing overhead costs at each step. To illustrate the approach, we consider the application of Hamiltonian simulation for which QSP implements a polynomial approximation of the time evolution operator. We test the protocol by running the algorithm on the Quantinuum H1-1 trapped-ion quantum computer powered by Honeywell. In particular, we compute the time dependence of bipartite entanglement entropies for Ising spin chains and find good agreements with exact numerical simulations. To make the best use of the device, we determine optimal experimental parameters by using a simplified error model for the hardware and numerically studying the trade-off between Hamiltonian simulation time, polynomial degree, and total accuracy. Our results are the first step in the experimental realization of QSP-based quantum algorithms.
We present the problem of approximating the time-evolution operator e-iHt to error ϵ, where the Hamiltonian H = ((G|⊗I)U (|Gi⊗I) is the projection of a unitary oracle U onto the state |Gi created by another unitary oracle. Our algorithm solves this with a query complexity O ( t + log(1/ϵ) ) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H is a density matrix. A key technical result is 'qubitization', which uses the controlled version of these oracles to embed any H in an invariant SU(2) subspace. A large class of operator functions of H can then be computed with optimal query complexity, of which e-iHt is a special case.