Abstract Quantum chemistry simulations on a quantum computer suffer from the overhead needed for encoding the Fermionic problem in a system of qubits. By exploiting the block diagonality of a Fermionic Hamiltonian, we show that the number of required qubits can be reduced while the number of terms in the Hamiltonian will increase. All operations for this reduction can be performed in operator space. The scheme is conceived as a pre-computational step that would be performed prior to the actual quantum simulation. We apply this scheme to reduce the number of qubits necessary to simulate both the Hamiltonian of the two-site Fermi–Hubbard model and the hydrogen molecule. Both quantum systems can then be simulated with a two-qubit quantum computer. Despite the increase in the number of Hamiltonian terms, the scheme still remains a useful tool to reduce the dimensionality of specific quantum systems for quantum simulators with a limited number of resources.
Abstract Quantum chemistry simulations on a quantum computer suffer from the overhead needed for encoding the Fermionic problem in a system of qubits. By exploiting the block diagonality of a Fermionic Hamiltonian, we show that the number of required qubits can be reduced while the number of terms in the Hamiltonian will increase. All operations for this reduction can be performed in operator space. The scheme is conceived as a pre-computational step that would be performed prior to the actual quantum simulation. We apply this scheme to reduce the number of qubits necessary to simulate both the Hamiltonian of the two-site Fermi–Hubbard model and the hydrogen molecule. Both quantum systems can then be simulated with a two-qubit quantum computer. Despite the increase in the number of Hamiltonian terms, the scheme still remains a useful tool to reduce the dimensionality of specific quantum systems for quantum simulators with a limited number of resources.
Since its introduction one decade ago, the quantum algorithm for chemistry has been among the most anticipated applications of quantum computers. However, as the age of industrial quantum technology dawns, so has the realization that even “polynomial” resource overheads are often prohibitive. There remains a large gap between the capabilities of existing hardware and the resources required to quantum compute classically intractable problems in chemistry. The primary contribution of this dissertation is to take meaningful steps towards reducing the costs of three approaches to quantum computing chemistry. First, we discuss how chemistry problems can be embedded in Hamiltonians suitable for commercially manufactured quantum annealing machines. We introduce schemes for more efficiently compiling problems to annealing Hamiltonians and apply the techniques to problems in protein folding, gene expression, and cheminformatics. Second, we introduce the first adiabatic quantum algorithm for fermionic simulation. Towards this end, we develop tools which embed arbitrary universal Hamiltonians in constrained hardware at a reduced cost. Finally, we turn our attention to the digital quantum algorithm for chemistry. By exploiting the locality of physical interactions, we quadratically reduce the number of terms which must be simulated. By analyzing the scaling of time discretization errors in terms of chemical properties, we obtain significantly tighter bounds on the minimum number of time steps which must be simulated. Also included in this dissertation is a protocol for preparing configuration interaction states that is asymptotically superior to all prior results and the details of the most accurate experimental quantum simulation of chemistry ever performed.
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 Rz and T 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.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
Model Hamiltonians described by fermionic operators are ubiquitous in physics and chemistry, but the quantum simulation of their properties is challenging on near-term quantum computers. Indeed, for any problem with more than two connected neighbors, standard mappings of fermions on qubits lead to many-body operators whose size increases with the size of the problem, quickly making the simulation unfeasible on noisy devices. Here we consider an alternative mapping based on qudits and the auxiliary fermion method, which leads to a significant reduction in the circuit depth and in the number of computational units to be controlled. These advantages increase for highly connected target Hamiltonians, such as lattices with long-range interactions or the molecular systems investigated in quantum chemistry. We numerically test our approach on qudits encoded in molecular spins, by performing the quantum simulation of a fermionic model on a small cluster. Published by the American Physical Society 2024
A potential approach for demonstrating quantum advantage is using quantum computers to simulate fermionic systems. Quantum algorithms for fermionic system simulation usually involve the Hamiltonian evolution and measurements. However, in the second quantization representation, the number of terms in many fermion system Hamiltonians, such as molecular Hamiltonians, is substantial, approximately O(N4), where N is the number of molecular orbitals. Due to this, the computational resources required for Hamiltonian evolution and expectation value measurements could be excessively large. To address this, we introduce a grouping strategy that partitions these O(N4) Hamiltonian terms into O(N2) groups, with the terms in each group mutually commuting. Based on this grouping method, we propose a parallel Hamiltonian evolution scheme that reduces the circuit depth of Hamiltonian evolution by a factor of N. Moreover, our grouping measurement strategy reduces the number of measurements needed to O(N2), whereas the current best grouping measurement schemes require O(N3) measurements. Additionally, we find that measuring the expectation value of a group of Hamiltonian terms requires fewer repetitions than measuring a single term individually, thereby reducing the number of quantum circuit executions. Our approach saves a factor of N3 in the overall time for Hamiltonian evolution and measurements, significantly decreasing the time required for quantum computers to simulate fermionic systems.
Simulating electronic structure on a quantum computer requires encoding of fermionic systems onto qubits. Common encoding methods transform a fermionic system of N spin-orbitals into an N-qubit system, but many of the fermionic configurations do not respect the required conditions and symmetries of the system so the qubit Hilbert space in this case may have unphysical states and thus cannot be fully utilized. We propose a generalized qubit-efficient encoding (QEE) scheme that requires the qubit number to be only logarithmic in the number of configurations that satisfy the required conditions and symmetries. For the case of considering only the particle-conserving and singlet configurations, we reduce the qubit count to an upper bound of O(mlog2N), where m is the number of particles. This QEE scheme is demonstrated on an H2 molecule in the 6-31G basis set and a LiH molecule in the STO-3G basis set using fewer qubits than the common encoding methods. We calculate the ground-state energy surfaces using a variational quantum eigensolver algorithm with a hardware-efficient ansatz circuit. We choose to use a hardware-efficient ansatz since most of the Hilbert space in our scheme is spanned by desired configurations so a heuristic search for an eigenstate is sensible. The simulations are performed on IBM Quantum machines and the Qiskit simulator with a noise model implemented from a IBM Quantum machine. Using the methods of measurement error mitigation and error-free linear extrapolation, we demonstrate that most of the distributions of the extrapolated energies using our QEE scheme agree with the exact results obtained by Hamiltonian diagonalization in the given basis sets within chemical accuracy. Our proposed scheme and results show the feasibility of quantum simulations for larger molecular systems in the noisy intermediate-scale quantum (NISQ) era. The number of terms in the Hamiltonian has an upper bound of $\mathcal{O}(\frac{{N}^{2m+1}}{(m\ensuremath{-}1)!\phantom{\rule{0.16em}{0ex}}m!})$ for the QEE scheme while it scales as O(N4) for the Jordan-Wigner encoding scheme. Nevertheless, we present several cases where QEE is useful.