Improved Hamiltonians for Quantum Simulations

Quantum simulations of lattice gauge theories for the foreseeable future will be hampered by limited resources. The historical success of improved lattice actions in classical simulations strongly suggests that Hamiltonians with improved discretization errors will reduce quantum resources, i.e. require $\gtrsim 2^d$ fewer qubits in quantum simulations for lattices with $d$ spatial dimensions. In this work, we consider $\mathcal{O}(a^2)$-improved Hamiltonians for pure gauge theories and design the corresponding quantum circuits for its real-time evolution in terms of primitive gates. An explicit demonstration for $\mathbb{Z}_2$ gauge theory is presented including exploratory tests using the ibm_perth device.

Introduction -Monte Carlo methods in lattice gauge theory (LGT), though powerful in many nonperturbative calculations, can suffer from sign problems -the Boltzmann weight during sampling becomes complex-valuedwhen simulating real-time dynamics. Thus, exponential resources are required to solve many interesting problems in particle physics, such as out-of-equilibrium evolution in the early universe [1], parton distribution function in hadron collisions [2][3][4], and the shear viscosity of the quark-gluon plasma [5]. Quantum computers can directly perform real-time simulations, avoiding these exponentially large resources plaguing classical methods [6][7][8]. Quantum simulation in the Hamiltonian formalism evolves the system with the time evolution operatorÛ(t) = e −iĤt . A HamiltonianĤ is constructed at finite lattice spacing a, causing discretization errors compared to the continuum theory in powers of a. Hamiltonians with discretizations scaling with lower powers of a require smaller lattice spacings for the same errors. This implies larger qubit requirements since the number of qubits is O((L/a) d ) for a d spatial dimensional lattice of length L.
The lattice gauge degrees of freedom, e.g. photons and gluons, need to be rendered finite and mapped to qubits . Current estimates for representing SU (3) suggest ∼ 10 qubits per gluon link [11,22,[31][32][33][34][35]. Further exacerbating the demand for qubits is the current, noisy status of quantum computers due to, e.g. entanglement with the environment and imperfect evolution. Though it remains an open question of how much quantum error correction is required to perform lattice simulations, general estimates suggest O(10 1−5 ) physical qubits per logical qubit [36][37][38] -so physical qubit requirements could easily rise to the megaqubyte scale for a 10 3 lattice.
The generically denseÛ(t) can only be efficiently con-structed approximately. For the decomposition in noncommuting termsĤ = iĤ i , a common approximation is [39,40]. Implementing U(t) for a LGT may require large number of quantum gates to achieve desirable precision. For example, in [34] a 10 3 lattice calculation of the shear viscosity η in QCD with errors of 10 −8 from trotterization and gate synthesis was estimated to require O(10 49 ) T gates -the most expensive gate for error-correcting quantum computers. Though these estimates could be reduced by considering only the low-lying states [41,42] or by relaxing the precision requirement to the level of uncertainties from lattice truncation, gate costs are still expected to be inaccessible in the near-term.
LGT specific error correction or mitigation could also decrease costs [56,57].
In this letter, we present a new direction for reducing quantum resources by using Hamiltonians with smaller discretization errors from finite differences. Quantum simulations can then be done at larger a, reducing the O((L/a) d ) qubits needed. We start with illustrating how to improve the commonly-used Kogut-Susskind Hamiltonian H KS [58] in the Symanzik improvement program [59][60][61], then derive time-evolution operators for the improved terms and construct the corresponding quantum circuits, followed by an explicit demonstration for Z 2 .
Improved Hamiltonians -For pure gauge theories, the classical Yang-Mills Hamiltonian can be written: where E(x) and B(x) are the electric and magnetic field strengths with spatial components E i (x) and B i (x). Al-arXiv:2203.02823v2 [hep-lat] 22 Aug 2022 ternatively, the magnetic energy density can be written in terms of F ij (x), the spatial-spatial field strength tensor, as: with Latin indices indicating spatial directions as shown in Fig. 1. In terms of color components, with λ b being generators of the gauge group. To ensure gauge invariance, lattice Hamiltonians are built from gauge links U i (x) = e igaAi(x) connecting lattice site x to its neighbor in the i spatial direction, with g being the gauge coupling and A i (x) the lattice gauge field [62]. By replacing the magnetic field B i (x) term with the plaquettes P ij (x) (see Fig. 1 for i = x and j = y) built from U i (x), and the electric field E i (x) with the lattice electric field L i (x), one arrives at H KS [58]: Re Tr P ij (x).
As temporal and spatial directions are treated differently, coupling g t and g s are introduced for the kinetic term K KS and potential term V KS , respectively. The discrepancy between H KS and H co is of O(a 2 ), as seen by seriesexpanding P ij with D i denoting the covariant derivative: For Symanzik improvement, one adds terms to H KS , and adjusts couplings to cancel the discretization errors [63,64]. The above classical O(a 2 ) error from F ij (D 2 i + D 2 j )F ij can be cancelled by including the rectangle term R ij (x) (see Fig. 1), as detailed in the Supplementary Material. At the quantum level O(g 2 s a 2 ) errors arise, requiring more terms, say the six-link bent loop terms C ijk (x) (see Fig. 1).
1: 3d lattice with example contributions to H I : the plaquette P xy , rectangles R yz and R zx , and the bent loop C xyz , and the two links U 1 and U 2 used for K 2L .
The improved Hamiltonian can be written as H I = [64]. V rect is defined as and V bent has analogous expressions to V rect . To cancel the O(a 2 ) errors in K KS , one adds the two-link term K 2L : For classical improvement, the couplings should be [63,64]: . Perturbative improvements at the quantum level generate corrections of O(g 2 a 2 ) [61,65]. One can further nonperturbatively tune these couplings numerically. For quantum simulations, these couplings could be extracted via analytic continuation of Euclidean calculations [55].  [66]. For the Lüscher-Weisz action, a = 0.4 fm lattices were found to have similar discretization errors to a = 0.17 fm lattices with the Wilson action [67]. Similar scaling is suggested by the limited direct studies of H I and H KS [68]. As the number of qubits required is O((L/a) d ), using H I may require 2 d fewer qubits in realistic quantum simulations for a fixed discretization error compared to H KS . While we occupy ourselves with pure gauge theory, future effort should consider the O(a) fermion Hamiltonians [69]particularly for chiral fermions.
Circuit Design -For quantum field theory calculations, H I is quantized by promoting the fields to operators: The magnetic field basis is the eigenbasis of the link operatorÛ while its Fourier transformation gives the electric field basis |L i diagonalizinĝ L 2 i . The quantum state of a link |U i is stored in a set of qubits -a link register. Any gauge circuit can be built from a set of primitive gates [70] acting on link registers: • left and right multiplication gates: withf denoting the Fourier transform of f .
• L-phase gate: U phase (θ) is a gauge group specific phase rotation, implemented by a diagonal matrix.
We implement the quantum circuits forĤ I term by term. Optimal quantum circuits depend on the underlying architecture -in particular connectivity. We assume register connectivity between a pair of links sharing a common site (linear register connectivity).
V I includesP ij (x) for every individual plaquette and the rectanglesR ij (x) for every neighboring two plaquettes. We denote the circuits forV KS as U V KS = e iθ Re TrPij (x) (Fig. 2a) and for the rectangles U Vrect = e iθ Re TrRij (x) (Fig. 2b), with the coupling and trotter step encoded in θ. The circuit of Fig. 2b with registers appropriately changed implements U V bent .
UV rect assuming linear register connectivity. The circuits U K KS = e iθ TrL 2 1 forK KS can be implemented by the L-phase gate U phase in the electric field basis [70], as shown in Fig. 2c. To avoid dealing withL andÛ operators simultaneously, we rewriteK 2L aŝ using the right electric field operator [19]: For simplicity, we denote the two succeeding links in one direction as U 1 and U 2 following Fig. 1 The circuit in Fig. 2d implements Eq. (9) by first storing the conserved quantity U 1 U 2 in the second register |U 2 via U L × , then performing e iθ TrL 2 1 on |U 1 with the sequence U † F U phase U F . Finally, we ensure the conserved product of While usingĤ I should require 2 d times fewer qubits, it requires additional gates to implement evolutions with the improved terms. Since the dominant quantum errors today are from decoherence and the entangling gates with error rates of O(10 −2 ) [71][72][73], this increased cost may diminish the gain from usingĤ I . We list the gate costs in terms of primitive gates in Tab. I for one trotter step using eitherĤ KS orĤ I . Depending on which primitive gates dominate the circuits, the gate cost forĤ I is 2 to 4 times that ofĤ KS per link register. For the group Z N and D N [74], different primitive gates take approximately the same order of entangling native gates. SinceĤ I should require 2 d fewer link registers, for the cases of d = 2, 3 we anticipate the same or fewer total primitive gate cost. Demonstration -For Z 2 gauge theory,Ĥ I can be mapped to Pauli matrices. Choosing the magnetic field basis, the qubit state |0 (|1 ) represents the element 1 (-1) of Z 2 . Implementations of the primitive gates are listed in the last column of Tab. I. We consider the most expensive Z 2 gate, U Vrect on the 7-qubit ibm perth device (Fig. 3c). The connectivity of ibm perth prevents implementing U Vrect as shown in Fig. 3a. With the mapping from links to qubits shown in Fig. 3c, a transpiled version of the circuit with 12 CNOTs and 20 additional one-qubit gates are used. We use the benchmark value θ = δt/(g s g t ) = 0.811411, precluding circuit optimization when using θ values such as π/2.
To quantify quantum errors, we evolve states with U Vrect and its inverse, and compare the measurement with quantum fidelity of U Vrect for the state |Ψ n =Ψ n |0 ⊗6 . Determining the fidelity requires testing all the possible states |Ψ n , a prohibitively expensive task [75]. Therefore we consider a restricted set consisting ofΨ n = m≤n H ⊗ m for n ∈ [0, 6] with m indicating the qubit to which H is applied.
To mitigate the coherent noise dominating the CNOT errors, we implement Pauli twirling [76][77][78][79][80] which converts coherent errors into random errors in Pauli channels and has found success in low-dimensional lattice field theories [81]. The circuits are modified by wrapping each CNOT with a set of Pauli gates {1, X, Y, Z} randomly sampled from sets satisfying where the i-th qubit (including spectators) was rotated by O(10) circuits to be sufficient for error mitigation [76]. Therefore we implemented 15 unique circuits and run each circuit 2 13 times. We also compute F |6 rect without Pauli twirling to gauge its effect.
With the above setup, we obtain the distribution P (w H ) in Fig. 5 for selected |Ψ n and the state-dependent fidelities F |n rect (Table II), yielding an average F rect = 0.550. Without Pauli twirling for n = 6, P (w H ) is indistinguishable from the noise-dominated limit while all the Pauli-twirled results are skewed toward the noiseless result, with states of lower n (and consequently less average entanglement) being closer to the desired value. Comparing the results for |Ψ 6 with and without Pauli twirling we observe a fourfold improvement in fidelity -clearly demonstrating the advantage from this error mitigation.  For a single trotter step, the time evolution ofĤ I for a two-plaquette lattice with open boundary conditions requires at least 28 CNOTs (40 one-qubit gates): 12 CNOTs (20 one-qubit gates) for U Vrect , at least 12 CNOTs (2 onequbit gates) for the two U V KS and 4 CNOTs (6 one-qubit gates) for the two U K 2L , alongwith 12 one-qubit gates for U K KS . Assuming that the average fidelity depends on the total number of CNOT gates, we can estimate the singletrotter-step fidelity forĤ I : F δ (F rect ) 28/12 ≈ 0.25. Thus current devices are inadequate for real-time computations. However given the expected hardware improvements in the coming years [36][37][38], F δ will be improved, allowing simulations of a two-plaquette lattice for Z 2 gauge theory and direct comparisons between Hamiltonians. Alternatively, classical simulators could explore lattices up to 7 2 [82] to test improved Hamiltonians.
In this letter, we designed the quantum circuits for simulating the improved HamiltonianĤ I . Comparing to the commonly usedĤ KS ,Ĥ I should allow quantum simulations with 2 d fewer qubits. With this reduction, we expect the gate count to be comparable or less than that ofĤ KS for theories with d ≥ 2 despite increases of gate costs per link. For near-term numerical demonstrations, we constructed the circuits forĤ I of the Z 2 gauge theory and found that for ibm perth the fidelity of the 12 CNOT improved potential term is 0.550. Our results suggest that alongside hardware developments, improved Hamiltonians can accelerate quantum simulations by years by reducing the number of qubits required, with optimistic prospects for 2 + 1d Z 2 simulations in the near future.
We would like to thank Erik Gustafson, Joseph Lykken and Michael Wagman for insightful discussions and comments on the manuscript. This work is supported by the Department of Energy through the Fermilab QuantiSED program in the area of "Intersections of QIS and Theoretical Particle Physics". Fermilab is operated by Fermi Research Alliance, LLC under contract number DE-AC02-07CH11359 with the United States Department of Energy. We acknowledge use of the IBM Q for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Q team.