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arXiv:2005.13070 · IEEE International Conference on Quantum Computing and Engineering (QCE)

On connectivity-dependent resource requirements for digital quantum simulation of d-level particles

Nicolas P. D. Sawaya, Gian Giacomo Guerreschi, Adam A. Holmes

Published Oct 1, 2020• 6 citations
arXivHTMLPDF

Abstract

A primary objective of quantum computation is to efficiently simulate quantum physics. Scientifically and technologically important quantum Hamiltonians include those with spin-s, vibrational, photonic, and other bosonic degrees of freedom, i.e. problems composed of or approximated by d-level particles (qudits). Recently, several methods for encoding these systems into a set of qubits have been introduced, where each encoding's efficiency was studied in terms of qubit and gate counts. Here, we build on previous results by including effects of hardware connectivity. To study the number of SWAP gates required to Trotterize commonly used quantum operators, we use both analytical arguments and automatic tools that optimize the schedule in multiple stages. We study the unary (or one-hot), Gray, standard binary, and block unary encodings, with three connectivities: linear array, ladder array, and square grid. Among other trends, we find that while the ladder array leads to substantial efficiencies over the linear array, the advantage of the square over the ladder array is less pronounced. These results are applicable in hardware co-design and in choosing efficient qudit encodings for a given set of near-term quantum hardware. Additionally, this work may be relevant to the scheduling of other quantum algorithms for which matrix exponentiation is a subroutine.

DOI
10.1109/qce49297.2020.00031

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Background: The structure and dynamics of many-body systems are the result of a delicate interplay between underlying interactions. Fermionic pairing, for example, plays a central role in various physical systems, ranging from condensed matter to nuclear systems, where it can lead to collective phenomena such as superconductivity and superfluidity. In atomic nuclei, the interplay between pairing and particle-hole interactions leads to a high degree of complexity and intricate entanglement structures. Despite this apparent complexity, symmetries emerge and manifest themselves in observable regular patterns. These symmetries and their breakings have long been used to determine relevant degrees of freedom and simplify classical descriptions of many-body systems.Purpose: This work explores the potential utility of quantum computers with arrays of qudits in simulating interacting fermionic systems, when the qudits can naturally map the relevant degrees of freedom determined by an underlying symmetry group.Method: The Agassi model of fermions interacting via particle-hole and pairing interactions is based on an underlying $\mathrm{so}(5)$ algebra. Such systems can intuitively be partitioned into pairs of modes with five basis states, which thus naturally map to arrays of $d=5$ qudits (qu5its). Classical noiseless simulations of the time evolution of systems with up to twelve qu5its are performed, by implementing quantum circuits that are developed herein, using python codes invoking Google's cirq software. The resource requirements of the qu5it circuits are analyzed and compared with two different mappings to qubit systems: a physics-aware Jordan-Wigner mapping requiring four qubits per mode pair and a state-to-state mapping requiring three qubits per mode pair.Results: While the dimensionality of Hilbert spaces in mappings to qu5it systems are less than those for the corresponding qubit systems, the number of entangling operations, depending on the available hardware, can either be greater or smaller than for the physics-aware Jordan-Wigner mapping. The state-to-state mapping, while having a smaller Hilbert space than Jordan-Wigner mappings, appears to be the least efficient in gate counts. Further, a previously unknown sign problem has been identified from Trotterization errors in time evolving high-energy excitations.Conclusions: There appear to be advantages in employing quantum computers with arrays of qudits to perform simulations of many-body dynamics that exploit the role of underlying symmetries, specifically in lowering the required quantum resources and in reducing anticipated errors that take the simulation out of the physical space. If the necessary entangling gates are not directly supported by the hardware, physics-aware mappings to qubits may, however, be advantageous for other aspects.

arXiv:2305.11941•Physical Review C•Dec 7, 2023•19 citations
A primary objective of quantum computation is to efficiently simulate quantum physics. Scientifically and technologically important quantum Hamiltonians include those with spin-s, vibrational, photonic, and other bosonic degrees of freedom, i.e. problems composed of or approximated by d-level particles (qudits). Recently, several methods for encoding these systems into a set of qubits have been introduced, where each encoding's efficiency was studied in terms of qubit and gate counts. Here, we build on previous results by including effects of hardware connectivity. To study the number of SWAP gates required to Trotterize commonly used quantum operators, we use both analytical arguments and automatic tools that optimize the schedule in multiple stages. We study the unary (or one-hot), Gray, standard binary, and block unary encodings, with three connectivities: linear array, ladder array, and square grid. Among other trends, we find that while the ladder array leads to substantial efficiencies over the linear array, the advantage of the square over the ladder array is less pronounced. These results are applicable in hardware co-design and in choosing efficient qudit encodings for a given set of near-term quantum hardware. Additionally, this work may be relevant to the scheduling of other quantum algorithms for which matrix exponentiation is a subroutine.
Abstract Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin- 12\frac{1}{2}21​ 1 2 systems. Here, we instead consider encodings of d -level (i.e., qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin- s and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity patterns, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common Hamiltonian classes used in physics and chemistry, while modeling the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to d , although clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers.
We compare the basic resource requirements for first and second quantized bosonic mappings in a system consisting of NNN particles in MMM modes. In addition to the standard binary first quantized mapping, we investigate the unary first quantized mapping. Our comparison focuses on the kkk-body reduced density matrix (kkk-RDM) and two standard bosonic Hamiltonians. The first quantized mappings use less resources for off-diagonal terms of the kkk-RDM by a factor of $ \sim N^k$, compared to the second quantized mappings. The number of gates for the first quantized binary mapping increases faster with MMM compared to the other mappings. Nevertheless, a detailed numeric analysis reveals that the binary first quantized mapping still requires fewer gates than the binary and unary second quantized ones for realistic combinations of NNN and MMM, while requiring exponentially fewer qubits than the unary mappings. Additionally, the number of CNOT and Rz(φ)R_z(φ)Rz​(φ) gates necessary to express a single Trotter step of the Hamiltonian in the binary first quantized mapping is comparable to the (most efficient for a single Trotter step) unary first quantized one when M=2nM = 2^nM=2n for both the Bose-Hubbard model and the harmonic trap with short-range interactions. Additionally the binary mapping leads to lower one-norms than the unary mapping making it the overall most efficient choice for qubitization-based quantum phase estimation.
Abstract Quantum computing hardware is undergoing rapid development from proof-of-principle devices to scalable machines that could eventually challenge classical supercomputers on specific tasks. On platforms with local connectivity, the transition from one- to two-dimensional arrays of qubits is seen as a natural technological step to increase the density of computing power and to reduce the routing cost of limited connectivity. Here we map and schedule representative algorithmic workloads—the Quantum Fourier Transform (QFT) relevant to factoring, the Grover diffusion operator relevant to quantum search, and Jordan–Wigner parity rotations relevant to simulations of quantum chemistry and materials science—to qubit arrays with varying connectivity. In particular we investigate the impact of restricting the ideal all-to-all connectivity to a square grid, a ladder and a linear array of qubits. Our schedule for the QFT on a ladder results in running time close to that of a system with all-to-all connectivity. Our results suggest that some common quantum algorithm primitives can be optimized to have execution times on systems with limited connectivities, such as a ladder and linear array, that are competitive with systems that have all-to-all connectivity.
Simulating chemical dynamics is computationally challenging, especially for nonadiabatic dynamics, where numerically exact classical simulations scale exponentially with system size, becoming intractable for even small molecules. On quantum computers, chemical dynamics can be simulated efficiently using either universal, qubit-only devices or specialized mixed-qudit-boson (MQB) simulators, which natively host electronic and vibrational degrees of freedom. Here, we compare the quantum resources required for a qubit-only approach to achieve the same accuracy as an MQB device at simulating nonadiabatic molecular dynamics. We find that MQB simulations require orders-of-magnitude fewer quantum operations than qubit-only simulations, with a one-gate MQB circuit requiring a qubit-equivalent circuit volume of over 400,000 when simulating an isolated molecule, which increases to over ten million when environmental effects are included. These estimates assume perfect qubits and gates, and would increase by additional orders of magnitude if error correction were used for fault tolerance. When errors are small, the advantage of MQB simulators becomes even larger as system size increases. Our results highlight the enormous resource advantages of representing non-qubit chemical degrees of freedom natively, rather than encoding them into qubits.
Abstract As the effort to scale up existing quantum hardware proceeds, it becomes necessary to schedule quantum gates in a way that minimizes the number of operations. There are three constraints that have to be satisfied: the order or dependency of the quantum gates in the specific algorithm, the fact that any qubit may be involved in at most one gate at a time, and the restriction that two-qubit gates are implementable only between connected qubits. The last aspect implies that the compilation depends not only on the algorithm, but also on hardware properties like connectivity. Here we suggest a two-step approach in which logical gates are initially scheduled neglecting connectivity considerations, while routing operations are added at a later step in a way that minimizes their overhead. We rephrase the subtasks of gate scheduling in terms of graph problems like edge-coloring and maximum subgraph isomorphism. While this approach is general, we specialize to a one-dimensional array of qubits to propose a routing scheme that is minimal in the number of exchange operations. As a practical application, we schedule the quantum approximate optimization algorithm in a linear geometry and quantify the reduction in the number of gates and circuit depth that results from increasing the efficacy of the scheduling strategies.
Background: The structure and dynamics of many-body systems are the result of a delicate interplay between underlying interactions. Fermionic pairing, for example, plays a central role in various physical systems, ranging from condensed matter to nuclear systems, where it can lead to collective phenomena such as superconductivity and superfluidity. In atomic nuclei, the interplay between pairing and particle-hole interactions leads to a high degree of complexity and intricate entanglement structures. Despite this apparent complexity, symmetries emerge and manifest themselves in observable regular patterns. These symmetries and their breakings have long been used to determine relevant degrees of freedom and simplify classical descriptions of many-body systems.Purpose: This work explores the potential utility of quantum computers with arrays of qudits in simulating interacting fermionic systems, when the qudits can naturally map the relevant degrees of freedom determined by an underlying symmetry group.Method: The Agassi model of fermions interacting via particle-hole and pairing interactions is based on an underlying so(5)\mathrm{so}(5)so(5) algebra. Such systems can intuitively be partitioned into pairs of modes with five basis states, which thus naturally map to arrays of d=5d=5d=5 qudits (qu5its). Classical noiseless simulations of the time evolution of systems with up to twelve qu5its are performed, by implementing quantum circuits that are developed herein, using python codes invoking Google's cirq software. The resource requirements of the qu5it circuits are analyzed and compared with two different mappings to qubit systems: a physics-aware Jordan-Wigner mapping requiring four qubits per mode pair and a state-to-state mapping requiring three qubits per mode pair.Results: While the dimensionality of Hilbert spaces in mappings to qu5it systems are less than those for the corresponding qubit systems, the number of entangling operations, depending on the available hardware, can either be greater or smaller than for the physics-aware Jordan-Wigner mapping. The state-to-state mapping, while having a smaller Hilbert space than Jordan-Wigner mappings, appears to be the least efficient in gate counts. Further, a previously unknown sign problem has been identified from Trotterization errors in time evolving high-energy excitations.Conclusions: There appear to be advantages in employing quantum computers with arrays of qudits to perform simulations of many-body dynamics that exploit the role of underlying symmetries, specifically in lowering the required quantum resources and in reducing anticipated errors that take the simulation out of the physical space. If the necessary entangling gates are not directly supported by the hardware, physics-aware mappings to qubits may, however, be advantageous for other aspects.