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Simulating a fermionic system on a quantum computer requires encoding the anti-commuting fermionic variables into the operators acting on the qubit Hilbert space. The most familiar of which, the Jordan-Wigner transformation, encodes fermionic operators into non-local qubit operators. As non-local operators lead to a slower quantum simulation, recent works have proposed ways of encoding fermionic systems locally. In this work, we show that locality may in fact be too strict of a condition and the size of operators can be reduced by encoding the system quasi-locally. We give examples relevant to lattice models of condensed matter and systems relevant to quantum gravity such as SYK models. Further, we provide a general construction for designing codes to suit the problem and resources at hand and show how one particular class of quasi-local encodings can be thought of as arising from truncating the state preparation circuit of a local encoding. We end with a discussion of designing codes in the presence of device connectivity constraints.
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Simulation of fermionic many-body systems on a quantum computer requires a suitable encoding of fermionic degrees of freedom into qubits. Here we revisit the Superfast Encoding introduced by Kitaev and one of the authors. This encoding maps a target fermionic Hamiltonian with two-body interactions on a graph of degree $d$ to a qubit simulator Hamiltonian composed of Pauli operators of weight $O(d)$. A system of $m$ fermi modes gets mapped to $n=O(md)$ qubits. We propose Generalized Superfast Encodings (GSE) which require the same number of qubits as the original one but have more favorable properties. First, we describe a GSE such that the corresponding quantum code corrects any single-qubit error provided that the interaction graph has degree $dge 6$. In contrast, we prove that the original Superfast Encoding lacks the error correction property for $dle 6$. Secondly, we describe a GSE that reduces the Pauli weight of the simulator Hamiltonian from $O(d)$ to $O(log{d})$. The robustness against errors and a simplified structure of the simulator Hamiltonian offered by GSEs can make simulation of fermionic systems within the reach of near-term quantum devices. As an example, we apply the new encoding to the fermionic Hubbard model on a 2D lattice.
Simulating quantum many-body systems is a highly demanding task since the required resources grow exponentially with the dimension of the system. In the case of fermionic systems, this is even harder since nonlocal interactions emerge due to the antisymmetric character of the fermionic wave function. Here, we introduce a digital-analog quantum algorithm to simulate a wide class of fermionic Hamiltonians including the paradigmatic Fermi-Hubbard model. These digital-analog methods allow quantum algorithms to run beyond digit
The ability to simulate a fermionic system on a quantum computer is expected to revolutionize chemical engineering, materials design, nuclear physics, to name a few. Thus, optimizing the simulation circuits is of significance in harnessing the power of quantum computers. Here, we address this problem in two aspects. In the fault-tolerant regime, we optimize the $rzgate$ and $tgate$ gate counts along with the ancilla qubit counts required, assuming the use of a product-formula algorithm for implementation. We obtain a savings ratio of two in the gate counts and a savings ratio of eleven in the number of ancilla qubits required over the state of the art. In the pre-fault tolerant regime, we optimize the two-qubit gate counts, assuming the use of the variational quantum eigensolver (VQE) approach. Specific to the latter, we present a framework that enables bootstrapping the VQE progression towards the convergence of the ground-state energy of the fermionic system. This framework, based on perturbation theory, is capable of improving the energy estimate at each cycle of the VQE progression, by about a factor of three closer to the known ground-state energy compared to the standard VQE approach in the test-bed, classically-accessible system of the water molecule. The improved energy estimate in turn results in a commensurate level of savings of quantum resources, such as the number of qubits and quantum gates, required to be within a pre-specified tolerance from the known ground-state energy. We also explore a suite of generalized transformations of fermion to qubit operators and show that resource-requirement savings of up to more than $20%$, in small instances, is possible.
Simulating quantum physics with a device which itself is quantum mechanical, a notion Richard Feynman originated, would be an unparallelled computational resource. However, the universal quantum simulation of fermionic systems is daunting due to their particle statistics, and Feynman left as an open question whether it could be done, because of the need for non-local control. Here, we implement fermionic interactions with digital techniques in a superconducting circuit. Focusing on the Hubbard model, we perform time evolution with constant interactions as well as a dynamic phase transition with up to four fermionic modes encoded in four qubits. The implemented digital approach is universal and allows for the efficient simulation of fermions in arbitrary spatial dimensions. We use in excess of 300 single-qubit and two-qubit gates, and reach global fidelities which are limited by gate errors. This demonstration highlights the feasibility of the digital approach and opens a viable route towards analog-digital quantum simulation of interacting fermions and bosons in large-scale solid state systems.
Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the energy can be expressed as a functional of the two-electron reduced density matrix (2-RDM) where the 2-RDM is a unique functional of the unencoded $N$-qubit-particle wave function. Contrasts are made with current hardware-efficient methods. An application to computing the ground-state energy and 2-RDM of H$_{4}$ is presented.
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