14 March 2021
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Strongly Universal Hamiltonian Simulators

The “Analog quantum simulation” paradigm of quantum computing aims to develop simpler models of a complex quantum system while reproducing all the physical attributes of the system in the operational domain of interest, such as its spectrum or phase diagram. The main idea is to simulate a rather complex target Hamiltonian H using a simpler Hamiltonian H’ that can be more easily implemented on practical analog quantum-computational hardware. One of the advantages of analog quantum simulation is the expected lesser requirement of quantum error correction or precise controls. Hence, it is considered to be an important practical direction in the era of NISQ technology.

The concept of universality when seeking analog simulators is based on the existence of a Hamiltonian H’ in the family that can be used to simulate any local Hamiltonian H. General universal models such as spin-lattice model Hamiltonians can potentially be inefficient to simulate directly as they for example require interaction energy that scales exponentially with system size. This exponential scaling holds true if the original Hamiltonian has higher-dimensional, long-range, or even all-to-all interactions. In this work, the authors provide an efficient construction of these strongly universal families in which the required interaction energy and all other resources in the 2D simulator scale polynomially instead of exponentially. The scaling occurs in the size of the target Hamiltonian and precision parameters and is independent of the target’s connectivity. The work involves the conversion of the target Hamiltonian to a quantum phase estimation circuit embedded in 1D. This circuit is then mapped back to a low-degree simulating Hamiltonian, using the Feynman-Kitaev circuit to-Hamiltonian construction.

The authors extend this method to simulate any target Hamiltonian with a 1D or 2D Hamiltonian using some of the existing techniques in the literature. Combinations of techniques such as the quantum phase estimation algorithm and circuit-to-Hamiltonian transformations were used in a non-perturbative way, which allows to overcome the exponential overhead common to previous constructions. The results show that only polynomial overheads in both particle number and interaction energy are sufficient to simulate any local Hamiltonian with arbitrary connectivity by some universal Hamiltonians embedded in 1D or 2D.

This work establishes the possibility of efficient universal analog quantum simulation using simple 1D or 2D systems, which we know can be built in practice with good control. The required constructions known so-far have been far from optimal. For example, existing hardware has limited types of interactions available, so in order to consider also general interactions, these need to be simulated using those single type of interactions together with ancilla qubits placed in more than one dimension. Polynomial-sized encoding and decoding circuits can be used to simulate 1D analog Hamiltonians which can be explored further towards achieving strong universality. In this work, it is shown that strongly universal analog quantum simulation is possible and can efficiently simulate any target Hamiltonian using 1D and 2D universal systems using polynomial qubits and interaction energies, which they show is tight since it is impossible to lower interaction energy to constant. However, the encoding circuits inducing non-local correlations can affect the desirable properties of analog Hamiltonian simulations such as preservation of locality of observables, as well as considerations of noise. As an alternative approach, the translation-invariance can be relaxed by letting Hamiltonian interactions have more free parameters to encode the target Hamiltonian.

One interesting takeaway from this research is that analog quantum simulation is actually relevant for many more systems than previously thought, and digital gate-based quantum simulation may not always be the best way to go in the described cases. Further experimental realizations of analog quantum simulators are required to develop methods to simulate all physical systems and tackle classically intractable problems in a practical and efficient way.

The concept of universality when seeking analog simulators is based on the existence of a Hamiltonian H’ in the family that can be used to simulate any local Hamiltonian H. General universal models such as spin-lattice model Hamiltonians can potentially be inefficient to simulate directly as they for example require interaction energy that scales exponentially with system size. This exponential scaling holds true if the original Hamiltonian has higher-dimensional, long-range, or even all-to-all interactions. In this work, the authors provide an efficient construction of these strongly universal families in which the required interaction energy and all other resources in the 2D simulator scale polynomially instead of exponentially. The scaling occurs in the size of the target Hamiltonian and precision parameters and is independent of the target’s connectivity. The work involves the conversion of the target Hamiltonian to a quantum phase estimation circuit embedded in 1D. This circuit is then mapped back to a low-degree simulating Hamiltonian, using the Feynman-Kitaev circuit to-Hamiltonian construction.

The authors extend this method to simulate any target Hamiltonian with a 1D or 2D Hamiltonian using some of the existing techniques in the literature. Combinations of techniques such as the quantum phase estimation algorithm and circuit-to-Hamiltonian transformations were used in a non-perturbative way, which allows to overcome the exponential overhead common to previous constructions. The results show that only polynomial overheads in both particle number and interaction energy are sufficient to simulate any local Hamiltonian with arbitrary connectivity by some universal Hamiltonians embedded in 1D or 2D.

This work establishes the possibility of efficient universal analog quantum simulation using simple 1D or 2D systems, which we know can be built in practice with good control. The required constructions known so-far have been far from optimal. For example, existing hardware has limited types of interactions available, so in order to consider also general interactions, these need to be simulated using those single type of interactions together with ancilla qubits placed in more than one dimension. Polynomial-sized encoding and decoding circuits can be used to simulate 1D analog Hamiltonians which can be explored further towards achieving strong universality. In this work, it is shown that strongly universal analog quantum simulation is possible and can efficiently simulate any target Hamiltonian using 1D and 2D universal systems using polynomial qubits and interaction energies, which they show is tight since it is impossible to lower interaction energy to constant. However, the encoding circuits inducing non-local correlations can affect the desirable properties of analog Hamiltonian simulations such as preservation of locality of observables, as well as considerations of noise. As an alternative approach, the translation-invariance can be relaxed by letting Hamiltonian interactions have more free parameters to encode the target Hamiltonian.

One interesting takeaway from this research is that analog quantum simulation is actually relevant for many more systems than previously thought, and digital gate-based quantum simulation may not always be the best way to go in the described cases. Further experimental realizations of analog quantum simulators are required to develop methods to simulate all physical systems and tackle classically intractable problems in a practical and efficient way.

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