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 $1-\delta$ 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 $\ln(1/\epsilon)$ , 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 $\ln(1/\delta)$ multiplicative factor to the query complexity, here we develop a new fully-coherent Hamiltonian simulation algorithm that achieves a query complexity additive in $\ln(1/\delta)$ : $\Theta\big( \|\mathcal{H}\| |t| + \ln(1/\epsilon) + \ln(1/\delta)\big)$ . 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.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 $1-\delta$ 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 $\ln(1/\epsilon)$, 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 $\ln(1/\delta)$ multiplicative factor to the query complexity, here we develop a new fully-coherent Hamiltonian simulation algorithm that achieves a query complexity additive in $\ln(1/\delta)$: $\Theta\big( \|\mathcal{H}\| |t| + \ln(1/\epsilon) + \ln(1/\delta)\big)$. 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.

Oct 21, 20211 citation

Quantum Algorithms for Hamiltonian Simulation

Barry C. Sanders, Graeme Ahokas, Richard Cleve +1 more

Chapman & Hall/CRC applied mathematics and nonlinear science seriesSep 19, 20070 citations

An efficient algorithm for approximate shadow Hamiltonian simulation

Abhijit Chakraborty, Bharath Sambasivam, Karunya Shirali +4 more

We propose an efficient algorithm based on shadow Hamiltonian simulation to approximately simulate the real-time dynamics of observables under time-independent Hamiltonians. Shadow Hamiltonian simulation works at the level of the operator algebra generated by the observables through commutators with the Hamiltonian. Exactly encoding the quantum state in this picture is generally inefficient for interacting systems due to the exponential growth of the operator algebra. Our algorithm overcomes this bottleneck by systematically identifying the elements of the algebra most relevant to the target observables. This targeted approach is a controlled approximation that yields a highly efficient quantum state encoding that substantially reduces the size of the qubit register required to perform the time evolution using the shadow Hamiltonian. We propose two main pruning schemes, one based on a predefined operator basis and another on a constructed Krylov basis. We also present a hybrid scheme that builds a Krylov basis within a pruned algebra in the predefined basis. We benchmark our algorithm using lattice spin systems in one and two dimensions, for both one- and higher-point correlators as observables.We propose an efficient algorithm based on shadow Hamiltonian simulation to approximately simulate the real-time dynamics of observables under time-independent Hamiltonians. Shadow Hamiltonian simulation works at the level of the operator algebra generated by the observables through commutators with the Hamiltonian. Exactly encoding the quantum state in this picture is generally inefficient for interacting systems due to the exponential growth of the operator algebra. Our algorithm overcomes this bottleneck by systematically identifying the elements of the algebra most relevant to the target observables. This targeted approach is a controlled approximation that yields a highly efficient quantum state encoding that substantially reduces the size of the qubit register required to perform the time evolution using the shadow Hamiltonian. We propose two main pruning schemes, one based on a predefined operator basis and another on a constructed Krylov basis. We also present a hybrid scheme that builds a Krylov basis within a pruned algebra in the predefined basis. We benchmark our algorithm using lattice spin systems in one and two dimensions, for both one- and higher-point correlators as observables.

quant-pharXiv:2607.11882Jul 14, 20260 citations

Efficient simulation of Hamiltonians

Robin Kothari

The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. \n \nIn the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. \n \nIn the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.

Aug 18, 20102 citations

Randomized Algorithms for Hamiltonian Simulation

Chi Zhang

Springer proceedings in mathematics & statisticsJan 1, 20128 citations

1-\delta

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/\epsilon)

, 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/\delta)

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/\delta)

\Theta\big( \|\mathcal{H}\| |t| + \ln(1/\epsilon) + \ln(1/\delta)\big)

. 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.