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 $\\mathcal{O}(N)$ qubits for an $N$ spin-orbital system, and the highly compressed first-quantized encoding requires $\\mathcal{O}(\\eta)$ qubits, where $\\eta<<N$ 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 $\\widetilde{\\cal O}(N^8t)$. 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 $\\widetilde{\\cal O}(N^5t)$. 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 $\\widetilde{\\cal O}(\\eta^2N^3t)$ 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.
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) qubits for an N spin-orbital system, and the highly compressed first-quantized encoding requires mathcalO(eta) qubits, where eta<<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). 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). 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) 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.
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.
We present a quantum algorithm for the simulation of molecular systems that is asymptotically more efficient than all previous algorithms in the literature in terms of the main problem parameters. As in Babbush et al (2016 New Journal of Physics 18 , 033032), we employ a recently developed technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The algorithm of this paper involves simulation under an oracle for the sparse, first-quantized representation of the molecular Hamiltonian known as the configuration interaction (CI) matrix. We construct and query the CI matrix oracle to allow for on-the-fly computation of molecular integrals in a way that is exponentially more efficient than classical numerical methods. Whereas second-quantized representations of the wavefunction require qubits, where N is the number of single-particle spin-orbitals, the CI matrix representation requires qubits, where is the number of electrons in the molecule of interest. We show that the gate count of our algorithm scales at most as .
We introduce novel algorithms for the quantum simulation of fermionic systems which are dramatically more efficient than those based on the Lie–Trotter–Suzuki decomposition. We present the first application of a general technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The key difficulty in applying algorithms for general sparse Hamiltonian simulation to fermionic simulation is that a query, corresponding to computation of an entry of the Hamiltonian, is costly to compute. This means that the gate complexity would be much higher than quantified by the query complexity. We solve this problem with a novel quantum algorithm for on-the-fly computation of integrals that is exponentially faster than classical sampling. While the approaches presented here are readily applicable to a wide class of fermionic models, we focus on quantum chemistry simulation in second quantization, perhaps the most studied application of Hamiltonian simulation. Our central result is an algorithm for simulating an N spin–orbital system that requires O ~ ( N 5 t ) gates. This approach is exponentially faster in the inverse precision and at least cubically faster in N than all previous approaches to chemistry simulation in the literature.
Quantum computation is one of the most promising new paradigms for the simulation of physical systems composed of electrons and atomic nuclei, with applications in chemistry, solid-state physics, materials science, and molecular biology. This requires a truncated representation of the electronic structure Hamiltonian using a finite number of orbitals. While it is, in principle, obvious how to improve on the representation by including more orbitals, this is usually unfeasible in practice (e.g., because of the limited number of qubits available) and severely compromises the accuracy of the obtained results. Here, we propose a quantum algorithm that improves on the representation of the physical problem by virtue of second-order perturbation theory. In particular, our quantum algorithm evaluates the second-order energy correction through a series of time-evolution steps under the unperturbed Hamiltonian. An important application is to go beyond the active-space approximation, allowing to include corrections of virtual orbitals, known as multireference perturbation theory. Here, we exploit the fact that the unperturbed Hamiltonian is diagonal for virtual orbitals and show that the number of qubits is independent of the number of virtual orbitals. This gives rise to more accurate energy estimates without increasing the number of qubits. Moreover, we demonstrate numerically for realistic chemical systems that the total runtime has highly favorable scaling in the number of virtual orbitals compared to previous studies. Numerical calculations confirm the necessity of the multireference perturbation theory energy corrections to reach accurate ground-state energy estimates. Our perturbation theory quantum algorithm can also be applied to symmetry-adapted perturbation theory. As such, we demonstrate that perturbation theory can help to reduce the quantum hardware requirements for quantum chemistry. Published by the American Physical Society 2024
In ab-initio quantum chemistry, solving the electronic structure problem of a molecular system of interest to obtain an approximation of the ground and, in some cases, the excited state energies is a core and central task in various quantum chemistry processes. These include, but certainly not limited to, the calculation of the activation energy–that is the minimal energy required to launch a given chemical reaction–, the dipole and multipole moments which describe the polarity of the molecule, performing the optimization of the geometry of the molecule and searching for transition states. Furthermore, many quantum chemistry applications are predicated, not merely on the approximation of the ground state energy, but also on the corresponding first order derivatives. The body of literature of classical algorithms approximating the ground state energy and its first order derivative is extensively rich with a plethora of methods such as self-consistent mean-field methods, coupled-cluster methods, density matrix renormalization group and many more, each of which has intrinsic set of properties and characteristics that makes it suitable for specific tasks and cases. However, classical methods face a practically insurmountable challenge: the efficient computation of high accuracy approximation of ground state energies of large system sizes. This is fundamentally because the electronic structure problem provably belongs to the complexity class of QMA-hard problems. Which also means that it is out of reach for quantum algorithms as well in general. However, with certain assumptions that are expected to hold molecules of interest–that so far proved to be classically challenging because the required accuracy set by the standard of chemical accuracy threshold (generally held in quantum chemistry)– quantum algorithms can efficiently treat such cases. While originally theorized by Richard Feynmann as a potentially efficient way of simulating some classically intractable many-body systems, quantum computing has since proven to have a stronger and richer potential paving the way for the inception of new algorithms such as Shor’s algorithm for integer factorization, HHL for solving linear system of equation and Grover’s algorithm for constrained binary search, just to name a few. These algorithms provably exhibit an improvement rate that goes respectively exponentially and quadratically in the time complexity. Quantum chemistry is considered a prominent candidate to benefit from quantum computing given the variety of quantum algorithms put forth to solve the electronic structure problem of particular systems of interest. These quantum algorithm proposals can be broadly categorized in three main categories that span the spectrum: from noisy intermediate scale quantum (NISQ) computers to early and full scale fault tolerant quantum ((E)FTQC) computers. Firstly, the variational quantum eigensolver (VQE) family of quantum algorithms which variationally search for the ground state energy through the optimization of the energy of a parametrized state prepared in a quantum computer using an ansatz. Secondly, the quantum Krylov subspace diagonalization (QKSD) methods, which generate a subspace of the original Hilbert space and efficiently produce the lowest energy in said subspace as its output. Lastly, we have the quantum phase estimation (QPE) algorithms, which provably produce the ground state energy in Heisenberg scaling presenting hence a provable efficient solution. In this thesis, we firstly set to work on the problem of efficient Hamiltonian linear combination of unitaries (LCU) representation. This is mainly motivated by the wide-range impact an efficient compression of the Hamiltonian can potentially have on several quantum algorithms such as VQE, QPE...etc. We designed a method that efficiently compress the Hamiltonian representation in a way that is beneficially impactful across the set of the aforementioned quantum algorithms. Namely, we motivate and present in Sections 1.3 and 1.4 respectively, the regularized compressed double-factorization (RC-DF) and show how it reduces the energy measurement complexity both in terms of number of distinct observables and number of samples. Moreover, it also reduces the total depth of the quantum phase estimation circuit thanks to its low normalization factor. Secondly, given the importance of the energy derivatives, in Sections 2.1 and 2.2, we introduce and present an efficient quantum algorithm for the calculation of the analytic gradient of VQE energies of double-factorized Hamiltonians. Furthermore, in Section 2.1, we expand on the utility of RC-DF by presenting a quantum algorithm that determines the analytic gradient of energies measured with respect to RC-DF Hamiltonains. Third, in Section 3.1, we summarize the theoretical results of purification error-mitigation techniques, namely echo verification and virtual distillation. In Section 3.2, we report on the results of the experimental bench mark, of said techniques, performed on on Google’s Sycamore device to validate the theoretical scaling of error suppression and extrapolate the resources needed for classically intractable system sizes. Fourth, in Section 4 we establish the challenges of computing energy gradients in a quantum Krylov frame work. To overcome the problem, we provide, in Section 4.3, an efficient quantum algorithm for calculating energy derivative obtained through QKSD procedures. Our approach shows how the QKSD ground state can be prepared using quantum signal processing (QSP) and reduces the number of observables to O(1) per reduced density matrix (RDM). Lastly, in light of the importance of quantum RAMs due to their utility in a variety of quantum algorithms that require efficient initial state preparation, as explained in Section 5.5, we address the problematic exponential depth of the bucket-brigade QRAM (when decomposed). We present in Section 5.6 a very efficient Clifford+T decomposition of the bucket-brigade QRAM, which preserves the linear scaling of the depth of the circuit without introducing any additional ancilla.
Abstract The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the O(N4) O ( N 4 ) gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with O(N3) O ( N 3 ) gate complexity in small simulations, which reduces to O(N2) O ( N 2 ) gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with O(N3) O ( N 3 ) gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have O(N2) O ( N 2 ) depth on a linearly connected array, an improvement over the O(N3) O ( N 3 ) scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 10 5 non-Clifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.