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/ ), 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/ ) + 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.

We extend molecular bootstrap embedding to make it appropriate for implementation on a quantum computer. This enables solution of the electronic structure problem of a large molecule as an optimization problem for a composite Lagrangian governing fragments of the total system, in such a way that fragment solutions can harness the capabilities of quantum computers. By employing state-of-art quantum subroutines including the quantum SWAP test and quantum amplitude amplification, we show how a quadratic speedup can be obtained over the classical algorithm, in principle. Utilization of quantum computation also allows the algorithm to match�at little additional computational cost�full density matrices at fragment boundaries, instead of being limited to 1-RDMs. Current quantum computers are small, but quantum bootstrap embedding provides a potentially generalizable strategy for harnessing such small machines through quantum fragment matching.

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: [Formula: see text]. 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.