Quantum dynamics simulations (QDSs) are one of the most highly anticipated applications of quantum computing. Quantum circuit depth for implementing Hamiltonian simulation algorithms is commonly time dependent so that long time dynamics simulations become impratical on near-term quantum processors. The Hamiltonian simulation algorithm based on Cartan decomposition (CD) provides an appealing scheme for QDSs with fixed-depth circuits while limited to time-independent case. In this work, we generalize this CD-based Hamiltonian simulation algorithm for studying time-dependent systems by combining it with variational Hamiltonian simulation. The time-dependent and time-independent parts of the Hamiltonian are treated with the variational approach and the CD-based Hamiltonian simulation algorithms, respectively. As such, only fixed-depth quantum circuits are required in this hybrid Hamiltonian simulation algorithm while still maintaining high accuracy. We apply this new algorithm to study the response of spin and molecular systems to $δ$ -kick electric fields and obtain accurate spectra for these excitation processes.Quantum dynamics simulations (QDSs) are one of the most highly anticipated applications of quantum computing. Quantum circuit depth for implementing Hamiltonian simulation algorithms is commonly time dependent so that long time dynamics simulations become impratical on near-term quantum processors. The Hamiltonian simulation algorithm based on Cartan decomposition (CD) provides an appealing scheme for QDSs with fixed-depth circuits while limited to time-independent case. In this work, we generalize this CD-based Hamiltonian simulation algorithm for studying time-dependent systems by combining it with variational Hamiltonian simulation. The time-dependent and time-independent parts of the Hamiltonian are treated with the variational approach and the CD-based Hamiltonian simulation algorithms, respectively. As such, only fixed-depth quantum circuits are required in this hybrid Hamiltonian simulation algorithm while still maintaining high accuracy. We apply this new algorithm to study the response of spin and molecular systems to $δ$-kick electric fields and obtain accurate spectra for these excitation processes.

arXiv:2406.14127Jun 20, 20240 citations

Efficient Fully-Coherent Hamiltonian Simulation

John M. Martyn, Yuan Liu, Zachary E. Chin +1 more

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

Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

Dominic William Berry, Graeme Ahokas, Richard Cleve +1 more

arXiv:quant-ph/0508139Communications in Mathematical PhysicsDec 13, 2006692 citations

Hybrid Hamiltonian Simulation for Excitation Dynamics

Lingyun Wan, Jie Liu, Zhenyu Li +1 more

Hamiltonian simulation is one of the most anticipated applications of quantum computing. Quantum circuit depth for implementing Hamiltonian simulation is commonly time dependent using Trotter-Suzuki product formulas so that long time quantum dynamic simulations (QDSs) become impratical for near-term quantum processors. Hamiltonian simulation based on Cartan decomposition (CD) provides an appealing scheme for QDSs with fixed-depth circuits, while it is limited to a time-independent Hamiltonian. In this work, we generalize this CD-based Hamiltonian simulation algorithm for studying time-dependent systems by combining it with variational quantum algorithms. The time-dependent and time-independent parts of the Hamiltonian are treated by using variational and CD-based Hamiltonian simulation algorithms, respectively. As such, this hybrid Hamiltonian simulation requires only fixed-depth quantum circuits to handle time-dependent cases while maintaining a high accuracy. We apply this new algorithm to study the response of spin and molecular systems to δ-kick electric fields and obtain accurate spectra for these excitation processes.Hamiltonian simulation is one of the most anticipated applications of quantum computing. Quantum circuit depth for implementing Hamiltonian simulation is commonly time dependent using Trotter-Suzuki product formulas so that long time quantum dynamic simulations (QDSs) become impratical for near-term quantum processors. Hamiltonian simulation based on Cartan decomposition (CD) provides an appealing scheme for QDSs with fixed-depth circuits, while it is limited to a time-independent Hamiltonian. In this work, we generalize this CD-based Hamiltonian simulation algorithm for studying time-dependent systems by combining it with variational quantum algorithms. The time-dependent and time-independent parts of the Hamiltonian are treated by using variational and CD-based Hamiltonian simulation algorithms, respectively. As such, this hybrid Hamiltonian simulation requires only fixed-depth quantum circuits to handle time-dependent cases while maintaining a high accuracy. We apply this new algorithm to study the response of spin and molecular systems to δ-kick electric fields and obtain accurate spectra for these excitation processes.

The Journal of Physical Chemistry LettersNov 1, 20243 citations

Quantum algorithms for quantum dynamics

Alexander Miessen, Pauline J. Ollitrault, Francesco Tacchino +1 more

Nature Computational ScienceDec 30, 202277 citations

δ

-kick electric fields and obtain accurate spectra for these excitation processes.

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.