Abstract This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from $$\mathcal {O}(2^n)$$ O ( 2 n ) to $$\mathcal {O}\left( \sum _{i=1}^{r} {} ^nC_r \right)$$ O ∑ i = 1 r n C r (for some $$r\ll n$$ r ≪ n ). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation.
Abstract This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from O(2n) O ( 2 n ) to O(∑i=1rnCr) O ∑ i = 1 r n C r (for some r≪n r ≪ n ). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation.
The exploration of potential energy operators in quantum systems holds paramount significance, offering profound insights into atomic behaviour, defining interactions, and enabling precise prediction of molecular dynamics. By embracing the Born-Oppenheimer picture, we delve into the intricate quantum evolution due to potential energy, facilitating accurate modelling and simulation of atomic phenomena with improved quantum fidelity. This research delves into time evolution operation due to potential energy functions for applications spanning quantum chemistry and condensed matter physics. Challenges in practical implementation, encompassing the formidable curse of dimensionality and intricate entangled interactions, are thoughtfully examined. Drawing upon seminal works, we lay a robust foundation for comprehensive investigations into potential energy landscapes with two proposed algorithms. In one methodology, we have shown a systematic decomposition of the potential energy function into Hadamard bases with composite construction of Pauli-Z, identity and RZ gates which can construct the unitary time evolution operator corresponding to the potential energy with a very high fidelity. The other method is a trade-off between complexity and fidelity, where we propose a novel quantum framework that can reduce the gate complexity from Θ(2n) to Θ(nCr ) (for some r < n). The proposed quantum algorithms are capable of efficiently simulating potential energy operators. The algorithms were implemented in simulators and IBM quantum hardware to prove their efficacy
The time evolution operator plays a crucial role in the precise computation of chemical experiments on quantum computers and holds immense promise for advancing the fields of physical and computer sciences, with applications spanning quantum simulation and machine learning. However, the construction of large-scale quantum computers poses significant challenges, prompting the need for innovative and resource-efficient strategies. Traditional methods like phase estimation or variational algorithms come with certain limitations such as the use of classical optimization or complex quantum circuitry. One successful method is the Trotterization technique used for quantum simulation, specifically in atomic structure problems with a gate complexity of approximately O(n^2) for an n-qubit realization. In this work, we have proposed a new encoding method, namely quantum approximate time evolution (QATE) for the quantum implementation of the kinetic energy operator as a diagonal unitary operator considering the first quantization level. The theoretical foundations of our approach are discussed, and experimental results are obtained on an IBM quantum machine. Our proposed method offers gate complexity in sub-quadratic polynomial with qubit size n which is an improvement over previous work. Further, the fidelity improvement for the time evolution of the Gaussian wave packet has also been demonstrated.
Quantum simulations based on first-quantized approaches can offer lower space and gate complexities than second-quantized approaches. The gate requirements for these simulations can be further reduced by employing pseudopotentials, a technique widely used in classical quantum chemistry calculations to decrease the number of simulated electrons. However, implementing pseudopotentials in quantum circuits remains challenging due to their non-local nature. In this study, we proposed a method to efficiently implement non-local potentials using matrix product state (MPS) encoding. MPS encoding allows us to efficiently implement a Trotter step operator with a circuit depth of O(deg(Vloc)ndeg(Vloc)+8Lnnatomχ2+2n2), where deg(Vloc) is the degree of the polynomial local potential, natom is the number of atoms in the system, L is the maximum number of projector functions across all atoms, and n is the number of qubits used to represent the binary-encoded wave function in one-dimensional space. We demonstrated the effectiveness of this approach by applying it to a model of the ionization of a one-dimensional hydrogen atom driven by an ultrashort laser pulse. Our results showed that MPS encoding significantly reduces the required circuit depth compared to a direct unitary decomposition, providing a pathway for practical first-quantized simulations.
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.
Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.
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.