We construct quantum circuits that exactly encode the spectra of correlated electron models up to errors from rotation synthesis.By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity Oðλ=ϵÞ, where λ is an absolute sum of Hamiltonian coefficients and ϵ is the target precision.For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T-gate complexity O(N þ logð1=ϵÞ), where N is the number of orbitals in the basis.This scenario enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 =ϵ þ N 2 logð1=ϵÞ=ϵ).Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime.Compiling to surface code fault-tolerant gates and assuming per-gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.
We construct quantum circuits that exactly encode the spectra of correlated electron models up to errors from rotation synthesis.By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity Oðλ=ϵÞ, where λ is an absolute sum of Hamiltonian coefficients and ϵ is the target precision.For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T-gate complexity O(N þ logð1=ϵÞ), where N is the number of orbitals in the basis.This scenario enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 =ϵ þ N 2 logð1=ϵÞ=ϵ).Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime.Compiling to surface code fault-tolerant gates and assuming per-gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.
We describe quantum circuits with only O(N ) Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of N arbitrary (e.g., molecular) orbitals.With O(λ/ ) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where λ is the 1-norm of Hamiltonian coefficients and is the target precision.This is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis.Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can work only with orthogonal-basis functions diagonalizing the Coloumb operator (e.g., the plane-wave dual basis).Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a nonorthogonal basis defined by the THC factors.We then use qubitization to simulate the nonorthogonal THC Hamiltonian, in a fashion that avoids most complications of the nonorthogonal basis.We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work.In addition to having lower asymptotic scaling space-time volume, compilation of our algorithm for challenging finitesized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach.By laying out and optimizing the surface-code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under 4 days of runtime, assuming 1-μs cycle times and physical gate-error rates no worse than 0.1%.
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.
Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle OC, and a direct sampling method for the amplitude oracle OA with SELECT-SWAP architecture, we achieve a T count that scales as O~(L) with respect to the number of interaction terms L in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the η-particle subspace, thereby reducing the subnormalization factor from O(L) to O(L), and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.
Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O ~ ( N 3 / 2 λ ) T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.
We propose a method for constructing circuits for quantum phase estimation of a molecular Hamiltonian in quantum chemistry by using variational optimization of quantum circuits solely on classical computers. The circuit generates a quantum state which encodes the coefficients of the terms in the Hamiltonian as probability amplitudes and plays a crucial role in the state-of-the-art efficient implementations of quantum phase estimation. We employ the automatic quantum circuit encoding algorithm [Shirakawa , ] to construct circuits, which requires classical simulations of quantum circuits of O ( log N ) qubits with N being the number of qubits of the Hamiltonian. The generated circuits do not need any ancillary qubit. We demonstrate our method by investigating the number of T -gates of the obtained circuits for quantum chemistry Hamiltonians of various molecules, which shows a constant-factor reduction compared to previous approaches that do not use ancillary qubits. Since the number of available logical qubits and T gates will be limited at the early stage of the fault-tolerant quantum computing, the proposed method is particularly of use for performing the quantum phase estimation with such limited capability. Published by the American Physical Society 2024
Recently, sample-based quantum diagonalization (SQD) has emerged as a promising approach to compute ground and excited states of problem Hamiltonians.This method classically diagonalizes a Hamiltonian in a subspace that is spanned by samples obtained from a quantum computer. However, by its nature, SQD suffers from a fundamental sampling problem, as some basis states that are required for a targeted accuracy may only be sampled extremely rarely. To alleviate this limitation, we introduce the SQD-AA algorithm that combines SQD with amplitude amplification (AA). SQD-AA uses AA to sequentially reduce probabilities of already measured bitstrings, thus making the observation of new ones more likely. We observe a reduction in the total query complexity of more than a factor 100 for algebraically and exponentially decaying model distributions, and analytically show a quadratic advantage for the latter. Moreover, we evaluate real molecules in an early fault-tolerant scenario and compare SQD-AA to SQD and iterative quantum phase estimation (iQPE). For all considered examples, we observe the lowest total number of T-gates for SQD-AA while only requiring circuits that are 3-4 orders of magnitude shallower than those needed for iQPE. Given this substantial reduction in circuit depth compared to iQPE while saving 2 orders of magnitude in total runtime compared to SQD, we expect a significant regime in early fault-tolerance where SQD-AA runs feasibly, but iQPE circuits are too deep to execute confidently.