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arXiv:0902.1419 · Physical Review A

Efficient quantum algorithm for preparing molecular-system-like states on a quantum computer

Hefeng Wang, Sahel Ashhab, Franco Nori

Published Apr 28, 2009• 48 citations
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Abstract

We present an efficient quantum algorithm for preparing a pure state on a quantum computer, where the quantum state corresponds to that of a molecular system with a given number $m$ of electrons occupying a given number $n$ of spin orbitals. Each spin orbital is mapped to a qubit: the states $|1⟩$ and $|0⟩$ of the qubit represent, respectively, whether the spin orbital is occupied by an electron or not. To prepare a general state in the full Hilbert space of $n$ qubits, which is of dimension ${2}^{n}$, $O({2}^{n})$ controlled-NOT gates are needed, i.e., the number of gates scales exponentially with the number of qubits. We make use of the fact that the state to be prepared lies in a smaller Hilbert space, and we find an algorithm that requires at most $O({2}^{m+1}{n}^{m}/m!)$ gates, i.e., scales polynomially with the number of qubits $n$, provided $n⪢m$. The algorithm is simulated numerically for the cases of the hydrogen molecule and the water molecule. The numerical simulations show that when additional symmetries of the system are considered, the number of gates to prepare the state can be drastically reduced, in the examples considered in this paper, by several orders of magnitude, from the above estimate.

DOI
10.1103/physreva.79.042335

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We present an efficient quantum algorithm for preparing a pure state on a quantum computer, where the quantum state corresponds to that of a molecular system with a given number mmm of electrons occupying a given number nnn of spin orbitals. Each spin orbital is mapped to a qubit: the states ∣1⟩|1⟩∣1⟩ and ∣0⟩|0⟩∣0⟩ of the qubit represent, respectively, whether the spin orbital is occupied by an electron or not. To prepare a general state in the full Hilbert space of nnn qubits, which is of dimension 2n{2}^{n}2n, O(2n)O({2}^{n})O(2n) controlled-NOT gates are needed, i.e., the number of gates scales exponentially with the number of qubits. We make use of the fact that the state to be prepared lies in a smaller Hilbert space, and we find an algorithm that requires at most O(2m+1nm/m!)O({2}^{m+1}{n}^{m}/m!)O(2m+1nm/m!) gates, i.e., scales polynomially with the number of qubits nnn, provided n⪢mn⪢mn⪢m. The algorithm is simulated numerically for the cases of the hydrogen molecule and the water molecule. The numerical simulations show that when additional symmetries of the system are considered, the number of gates to prepare the state can be drastically reduced, in the examples considered in this paper, by several orders of magnitude, from the above estimate.
We present an efficient quantum algorithm for preparing a pure state on a quantum computer, where the quantum state corresponds to that of a molecular system with a given number mmm of electrons occupying a given number nnn of spin orbitals. Each spin orbital is mapped to a qubit: the states ∣1>| 1 >∣1> and ∣0>| 0>∣0> of the qubit represent, respectively, whether the spin orbital is occupied by an electron or not. To prepare a general state in the full Hilbert space of nnn qubits, which is of dimension 2n2^{n}2n%, O(2n)O(2^{n})O(2n) CNOT gates are needed, i.e. the number of gates scales \emph{% exponentially} with the number of qubits. We make use of the fact that the state to be prepared lies in a smaller Hilbert space, and we find an algorithm that requires at most O((2n)m/m!)O((2 n)^{m}/{m!})O((2n)m/m!) CNOT gates, i.e. scales \emph{polynomially} with the number of qubits nnn, provided n≫mn\gg mn≫m. The algorithm is simulated numerically to prepare the electronic states of the hydrogen molecule and the water molecule. We show that when additional symmetries of the system are considered, the number of gates to prepare the state can be reduced dramatically.
Simulating electronic structure on a quantum computer requires encoding of fermionic systems onto qubits. Common encoding methods transform a fermionic system of NNN spin-orbitals into an NNN-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)\mathcal{O}(m{log}_{2}N)O(mlog2​N), where mmm is the number of particles. This QEE scheme is demonstrated on an H2{\mathrm{H}}_{2}H2​ molecule in the 6-31G basis set and a LiH\mathrm{LiH}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)\mathcal{O}({N}^{4})O(N4) for the Jordan-Wigner encoding scheme. Nevertheless, we present several cases where QEE is useful.
The quantum simulation of real molecules and materials is one of the most highly anticipated applications of quantum computing. Algorithms for simulating electronic structure using a first-quantized plane-wave representation are especially promising due to their asymptotic efficiency. However, previous proposals for preparing initial states for these simulation algorithms scale poorly with the size of the basis set. We address this shortcoming by showing how to efficiently map states defined in a Gaussian-type orbital basis to a plane-wave basis with a scaling that is logarithmic in the number of plane waves. Our key technical result is a proof that molecular orbitals constructed from Gaussian-type basis functions can be compactly represented in a plane-wave basis using matrix product states. While we expect that other approaches could achieve the same logarithmic scaling with respect to basis-set size, our proposed state-preparation technique is also highly efficient in practice. For example, in a series of numerical experiments on small molecules, we find that our approach allows us to prepare an approximation to the Hartree-Fock state using orders of magnitude fewer non-Clifford gates than a naive approach. By resolving the issue of state preparation, our work allows for the first quantum simulation of molecular systems the end-to-end complexity of which is truly sublinear in the basis-set size.
The calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms. We demonstrate that such algorithms can be applied to problems of chemical interest using modest numbers of quantum bits. Calculations of the water and lithium hydride molecular ground-state energies have been carried out on a quantum computer simulator using a recursive phase-estimation algorithm. The recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4. Mappings of the molecular wave function to the quantum bits are described. An adiabatic method for the preparation of a good approximate ground-state wave function is described and demonstrated for a stretched hydrogen molecule. The number of quantum bits required scales linearly with the number of basis functions, and the number of gates required grows polynomially with the number of quantum bits.
In quantum chemistry, the "electronic structure problem" refers to the process of numerically solving Schrodinger's equation for a given molecular system. These calculations rank among the most CPU-intensive computations that are performed nowadays, yet they are also crucial to the design of new drugs and materials. The idea behind quantum simulation of quantum chemistry is that a quantum computer could perform such calculations in a manner that is exponentially more efficient than what a classical computer could accomplish. In fact, quantum simulation is quite possibly the application most naturally suited to a quantum computer. In this work we introduce novel algorithms for the simulation of quantum chemistry, algorithms that are exponentially more precise than previous algorithms in the literature. Previous algorithms are based on the Trotter-Suzuki decomposition and scale like the inverse of the desired precision, while our algorithms represent the first practical application of the recently developed truncated Taylor series approach, and they achieve scaling that is logarithmic in the inverse of the precision. We explore algorithms for both second-quantized and first-quantized encodings of the chemistry Hamiltonian, where the second-quantized encoding requires mathcalO(N)\\mathcal{O}(N)mathcalO(N) qubits for an NNN spin-orbital system, and the highly compressed first-quantized encoding requires mathcalO(eta)\\mathcal{O}(\\eta)mathcalO(eta) qubits, where eta<<N\\eta<<Neta<<N is the number of electrons in the system. We first introduce a second-quantized algorithm that uses pre-computed molecular integrals, resulting in a gate count that scales like widetildecalO(N8t)\\widetilde{\\cal O}(N^8t)widetildecalO(N8t). Next we introduce algorithms that compute molecular integrals on the fly by carefully discretizing these integrals. Indeed, figuring out how to evaluate the integrals in an efficient manner is one of the main challenges of applying the truncated Taylor series approach to the chemistry problem: the complexity of a general quantum algorithm is traditionally formulated in terms of the number of queries made to an oracle whose assumption we just assume, but in this case, we actually construct the oracle for our specific problem. The evaluation of the oracle itself requires evaluating molecular integrals, which adds additional complexity to an algorithm. We find that the gate count of the second-quantized on-the-fly algorithm scales like widetildecalO(N5t)\\widetilde{\\cal O}(N^5t)widetildecalO(N5t). We also find that the first-quantized on-the-fly algorithm, which uses the configuration interaction (CI) matrix representation of the Hamiltonian, requires at most widetildecalO(eta2N3t)\\widetilde{\\cal O}(\\eta^2N^3t)widetildecalO(eta2N3t) gates. In general, the discovery of exponentially more precise algorithms will allow for the simulation of larger molecular systems than was previously possible under Trotter-based methods. In addition, the results here can be generalized to the simulation of other fermionic systems beyond the electronic structure problem.
Abstract The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the H2{H}_{2}H2​ H 2 and LiHLiHLiH L i H molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.