arXiv subjects

Joel Klassen

Publications and source records attributed to Joel Klassen.

15 recordsLinked to original sources

arXiv2025

Fermionic dynamics on a trapped-ion quantum computer beyond exact classical simulation

Simulation of the time-dynamics of fermionic many-body systems has long been predicted to be one of the key applications of quantum computers. Such simulations -- for which classical methods are often inaccurate -- are critical to advancing our knowledge and understanding of quantum chemistry and materials, underpinning a wide range of fields, from biochemistry to clean-energy technologies and chemical synthesis. However, the performance of all previous digital quantum simulations of fermions has been matched by classical methods, and it has thus far remained unclear whether near-term, intermediate-scale quantum hardware could offer any computational advantage in this area. Here, we implement an efficient quantum simulation algorithm on Quantinuum's System Model H2 trapped-ion quantum computer for the time dynamics of a 56-qubit system that is too complex for exact classical simulation. We focus on the periodic spinful 2D Fermi-Hubbard model and present evidence of spin-charge separation, where the elementary electron's charge and spin decouple. In the limited cases where ground truth is available through exact classical simulation, we find that it agrees with the results we obtain from the quantum device. Employing long-range Wilson operators to study deconfinement of the effective gauge field between spinons and the effective potential between charge carriers, we find behaviour that differs from predictions made by classical tensor network methods. Our results herald the use of quantum computing for simulating strongly correlated electronic systems beyond the capacity of classical computing.

quant-ph↗

arXiv2025

Programmable digital quantum simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits

Simulating the time-dynamics of quantum many-body systems was the original use of quantum computers proposed by Feynman, motivated by the critical role of quantum interactions between electrons in the properties of materials and molecules. Accurately simulating such systems remains one of the most promising applications of general-purpose digital quantum computers, in which all the parameters of the model can be programmed and any desired physical quantity output. However, performing such simulations on today's quantum computers at a scale beyond the reach of classical methods requires advances in the efficiency of simulation algorithms and error mitigation techniques. Here we demonstrate programmable digital quantum simulation of the dynamics of the 2D Fermi-Hubbard model -- one of the best-known simplified models of electrons in crystalline solids -- at a scale beyond exact classical state-vector simulation. We implement simulations of this model on lattice sizes up to ${6\times 6}$ using 72 qubits on Google's Willow quantum processor, across a range of physical parameters, including different on-site electron-electron interaction strengths and magnetic flux values, and study phenomena including formation of magnetic polarons (charge carriers surrounded by local magnetic polarisation), dynamical symmetry-breaking in stripe-ordered states, attraction of charge carriers on an entangled background state known as a valence bond solid, and the approach to equilibrium through thermalisation. We validate our results against exact calculations in parameter regimes where these are feasible, and compare them to approximate classical simulations performed using tensor network and operator propagation methods. Our results demonstrate that meaningful programmable digital quantum simulation of many-body interacting electron models is now feasible on state-of-the-art quantum hardware.

quant-ph↗

arXiv2024

Quantum-centric Supercomputing for Materials Science: A Perspective on Challenges and Future Directions

Computational models are an essential tool for the design, characterization, and discovery of novel materials. Hard computational tasks in materials science stretch the limits of existing high-performance supercomputing centers, consuming much of their simulation, analysis, and data resources. Quantum computing, on the other hand, is an emerging technology with the potential to accelerate many of the computational tasks needed for materials science. In order to do that, the quantum technology must interact with conventional high-performance computing in several ways: approximate results validation, identification of hard problems, and synergies in quantum-centric supercomputing. In this paper, we provide a perspective on how quantum-centric supercomputing can help address critical computational problems in materials science, the challenges to face in order to solve representative use cases, and new suggested directions.

quant-ph↗

arXiv2022

Towards near-term quantum simulation of materials

Simulation of materials is one of the most promising applications of quantum computers. On near-term hardware the crucial constraint on these simulations is circuit depth. Many quantum simulation algorithms rely on a layer of unitary evolutions generated by each term in a Hamiltonian. This appears in time-dynamics as a single Trotter step, and in variational quantum eigensolvers under the Hamiltonian variational ansatz as a single ansatz layer. We present a new quantum algorithm design for materials modelling where the depth of a layer is independent of the system size. This design takes advantage of the locality of materials in the Wannier basis and employs a tailored fermionic encoding that preserves locality. We analyse the circuit costs of this approach and present a compiler that transforms density functional theory data into quantum circuit instructions -- connecting the physics of the material to the simulation circuit. The compiler automatically optimises circuits at multiple levels, from the base gate level to optimisations derived from the physics of the specific target material. We present numerical results for materials spanning a wide structural and technological range. Our results demonstrate a reduction of many orders of magnitude in circuit depth over standard prior methods that do not consider the structure of the Hamiltonian. For example our results improve resource requirements for Strontium Vanadate (SrVO$_3$) from 864 to 180 qubits for a $3\times3\times3$ lattice, and the circuit depth of a single Trotter or variational layer from $7.5\times 10^8$ to depth $884$. Although this is still beyond current hardware, our results show that materials simulation may be feasible on quantum computers without necessarily requiring scalable, fault-tolerant quantum computers, provided quantum algorithm design incorporates understanding of the materials and applications.

quant-ph↗

arXiv2022

Optimizing fermionic encodings for both Hamiltonian and hardware

In this work we present a method for generating a fermionic encoding tailored to a set of target fermionic operators and to a target hardware connectivity. Our method uses brute force search, over the space of all encodings which map from Majorana monomials to Pauli operators, to find an encoding which optimizes a target cost function. In contrast to earlier works in this direction, our method searches over an extremely broad class of encodings which subsumes all known second quantized encodings that constitute algebra homomorphisms. In order to search over this class, we give a clear mathematical explanation of how precisely it is characterized, and how to translate this characterization into constructive search criteria. A benefit of searching over this class is that our method is able to supply fairly general optimality guarantees on solutions. A second benefit is that our method is, in principal, capable of finding more efficient representations of fermionic systems when the set of fermionic operators under consideration are faithfully represented by a smaller quotient algebra. Given the high algorithmic cost of performing the search, we adapt our method to handle translationally invariant systems that can be described by a small unit cell that is less costly. We demonstrate our method on various pairings of target fermionic operators and hardware connectivities. We additionally show how our method can be extended to find error detecting fermionic encodings in this class.

quant-ph↗

arXiv2022

Termwise versus globally stoquastic local Hamiltonians: questions of complexity and sign-curing

We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We show that the stoquastic local Hamiltonian problem is $\textbf{StoqMA}$-complete even for globally stoquastic Hamiltonians. We study the complexity of deciding whether a local Hamiltonian is globally stoquastic or not. In particular, we prove $\textbf{coNP}$-hardness of deciding global stoquasticity in a fixed basis and $Σ_2^p$-hardness of deciding global stoquasticity under single-qubit transformations. As a last result, we expand the class of sign-curing transformations by showing how Clifford transformations can sign-cure a class of disordered 1D $XYZ$ Hamiltonians.

quant-ph↗

arXiv2021

Phase Estimation of Local Hamiltonians on NISQ Hardware

In this work we investigate a binned version of Quantum Phase Estimation (QPE) set out by [Somma 2019] and known as the Quantum Eigenvalue Estimation Problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, [Clinton et al 2020], can improve the performance of QEEP in the NISQ regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in [Clinton et al 2020]. Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, given a $10\%$ acceptable error on a $3\times 3$ spin Fermi-Hubbard model, with a depolarizing noise rate of $10^{-6}$, we find that the phase estimation protocol of [Somma 2019] could be performed with a bin width of approximately $1/9$ the total spectral range at the circuit depth where traditional gate synthesis methods would yield a bin width that covers the entire spectral range. We explore possible modifications to this protocol and propose an application, which we call Randomized Quantum Eigenvalue Estimation Problem (rQeep). rQeep outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians, such as spectral gaps.

quant-ph↗

arXiv2021

A Compact Fermion to Qubit Mapping Part 2: Alternative Lattice Geometries

In recent work [arXiv:2003.06939v2] a novel fermion to qubit mapping -- called the compact encoding -- was introduced which outperforms all previous local mappings in both the qubit to mode ratio, and the locality of mapped operators. There the encoding was demonstrated for square and hexagonal lattices. Here we present an extension of that work by illustrating how to apply the compact encoding to other regular lattices. We give constructions for variants of the compact encoding on all regular tilings with maximum degree 4. These constructions yield edge operators with Pauli weight at most 3 and use fewer than 1.67 qubits per fermionic mode. Additionally we demonstrate how the compact encoding may be applied to a cubic lattice, yielding edge operators with Pauli weight no greater than 4 and using approximately 2.5 qubits per mode. In order to properly analyse the compact encoding on these lattices a more general group theoretic framework is required, which we elaborate upon in this work. We expect this framework to find use in the design of fermion to qubit mappings more generally.

quant-ph↗

arXiv2020

A Compact Fermion to Qubit Mapping

Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a novel fermion to qubit mapping which outperforms all previous local mappings in both the qubit to mode ratio, and the locality of mapped operators.

quant-ph↗

arXiv2020

Mitigating Errors in Local Fermionic Encodings

Quantum simulations of fermionic many-body systems crucially rely on mappings from indistinguishable fermions to distinguishable qubits. The non-local structure of fermionic Fock space necessitates encodings that either map local fermionic operators to non-local qubit operators, or encode the fermionic representation in a long-range entangled code space. In this latter case, there is an unavoidable trade-off between two desirable properties of the encoding: low weight representations of local fermionic operators, and a high distance code space. Here it is argued that despite this fundamental limitation, fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly for two fermionic encodings: the Verstraete-Cirac encoding, and an encoding appearing in concurrent work by Derby and Klassen. In these encodings many, but not all, single-qubit errors can be detected. However we show that the remaining undetectable single-qubit errors map to local, low-weight fermionic phase noise. We argue that such noise is natural for fermionic lattice models. This suggests that even when employing low-weight fermionic encodings, error rates can be suppressed in a similar fashion to high distance codes, provided one is willing to accept simulated natural fermionic noise in their simulated fermionic system.

quant-ph↗

arXiv2020

Hardness and Ease of Curing the Sign Problem for Two-Local Qubit Hamiltonians

We examine the problem of determining whether a multi-qubit two-local Hamiltonian can be made stoquastic by single-qubit unitary transformations. We prove that when such a Hamiltonian contains one-local terms, then this task can be NP-hard. This is shown by constructing a class of Hamiltonians for which performing this task is equivalent to deciding $3$-SAT. In contrast, we show that when such a Hamiltonian contains no one-local terms then this task is easy, namely we present an algorithm which decides, in a number of arithmetic operations over $\mathbb{R}$ which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.

quant-ph↗

arXiv2019

Two-local qubit Hamiltonians: when are they stoquastic?

We examine the problem of determining if a 2-local Hamiltonian is stoquastic by local basis changes. We analyze this problem for two-qubit Hamiltonians, presenting some basic tools and giving a concrete example where using unitaries beyond Clifford rotations is required in order to decide stoquasticity. We report on simple results for $n$-qubit Hamiltonians with identical 2-local terms on bipartite graphs. Our most significant result is that we give an efficient algorithm to determine whether an arbitrary $n$-qubit XYZ Heisenberg Hamiltonian is stoquastic by local basis changes.

quant-ph↗

arXiv2016

Quantum state tomography via reduced density matrices

Quantum state tomography via local measurements is an efficient tool for characterizing quantum states. However it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results advance the project of performing efficient and accurate quantum state tomography in practice.

quant-ph↗

arXiv2015

Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking

We study a chain of ferromagnetic nano-particles or ferromagnetic molecule/atoms on a substrate of fully gapped superconductors. We find that under quite realistic conditions, the fermion-number-parity symmetry $Z_2^f$ can spontaneously break. In other words, such a chain can realize a 1+1D fermionic topologically ordered state and the corresponding two-fold topological degeneracy on an open chain. Such a topological degeneracy becomes the so called Majorana zero mode in the non-interacting limit. More specifically, we find that $Z_2^f$ symmetry breaking or fermionic 1+1D topological order can appear if (1) the electron hopping $t_{ij}$ between nano-particles is larger than the energy splitting $δE_{eo}$ between the ground states of even and odd electrons on a nano-particle, (2) the Josephson coupling $J_i$ between the superconducting substrate and the nano-particle is larger than or similar to $δE_{eo}$, and (3) the electron hopping amplitude $t_{ij}$ is complex, or more precisely, the phase of gauge invariant combination $J_i t_{ij}^2 J_j^*$ is not zero.

cond-mat.str-el↗

arXiv2013

Universal Entanglers for Bosonic and Fermionic Systems

A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$, a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq 8$). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.

quant-ph↗