Abstract
The quantum circuit model is the defacto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below” the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware and deriving analytic circuit identities for synthesising multiqubit evolutions from twoqubit interactions. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. To quantify the benefits of this approach, we apply it to timedynamics simulation of the 2D spin FermiHubbard model. Combined with new error bounds for Trotter product formulas tailored to the nonasymptotic regime and an analysis of error propagation, we find that e.g. for a 5 × 5 FermiHubbard lattice we reduce the circuit depth from 1, 243, 586 using the best previous fermion encoding and error bounds in the literature, to 3, 209 in the pergate error model, or the circuitdepthequivalent to 259 in the pertime error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for nontrivial examples, significantly closer to being feasible in the NISQ era.
Introduction
Quantum computing is on the cusp of entering the era in which quantum hardware can no longer be simulated effectively classically, even on the world’s biggest supercomputers^{1,2,3,4,5}. Google recently achieved the first socalled "quantum supremacy” milestone demonstrating this^{6}. While reaching this milestone is an impressive experimental physics achievement, the very definition of this goal allows it to be a demonstration that has no useful practical applications^{7}. The recent Google results are of exactly this nature. By far the most important question for quantum computing now is to determine whether there are useful applications of this class of noisy, intermediatescale quantum (NISQ) hardware^{8}.
However, current quantum hardware is still extremely limited, with ≈50 qubits capable of implementing quantum circuits up to a gate depth of ≈20^{6}. This is far too limited to run useful instances of even the simplest textbook quantum algorithms, let alone implement the error correction and fault tolerance required for largescale quantum computations. Estimates of the number of qubits and gates required to run Shor’s algorithm on integers that cannot readily be factored on classical computers place it—and related numbertheoretic algorithms—well into the regime of requiring a fully scalable, faulttolerant quantum computer^{9,10}. Studies of practically relevant combinatorial problems tell a similar story for capitalising on the quadratic speedup of Grover’s algorithm^{11}. Quantum computers are naturally well suited for simulation of quantum manybody systems^{12,13}—a task that is notoriously difficult on classical computers. Quantum simulation is likely to be one of the first practical applications of quantum computing. But, while the number of qubits required to run interesting quantum simulations may be lower than for other applications, careful studies of the gate counts required for a quantum chemistry simulation of molecules that are not easily tractible classically^{14}, or for simple condensed matter models^{15}, remain far beyond current hardware.
With severely resourceconstrained hardware such as this, squeezing every ounce of performance out of it is crucial. The quantum circuit model is the standard way to design quantum algorithms, and quantum gates and circuits provide a highly convenient abstraction of quantum hardware. Circuits sit at a significantly lower level of abstraction than even assembly code in classical computing. But any layer of abstraction sacrifices some overhead for the sake of convenience. The quantum circuit model is no exception.
In the underlying hardware, quantum gates are typically implemented by controlling interactions between qubits. E.g. by changing voltages to bring superconducting qubits in and out of resonance, or by laser pulses to manipulate the internal states of trapped ions. By restricting to a fixed set of standard gates, the circuit model abstracts away the full capabilities of the underlying hardware. In the NISQ era, it is not clear this sacrifice is justified. The SolovayKitaev theorem tells us that the overhead of any particular choice of universal gate set is at most polylogarithmic^{16,17}. But when the available circuit depth is limited to ≈20, even a constant factor improvement could make the difference between being able to run an algorithm on current hardware, and being beyond the reach of foreseeable hardware.
The advantages of designing quantum algorithms "one level below” the circuit model become particularly acute in the case of Hamiltonian timedynamics simulation. To simulate evolution under a manybody Hamiltonian H = ∑_{〈i, j〉}h_{ij}, the basic Trotterization algorithm^{13,18} repeatedly timeevolves the system under each individual interaction h_{ij} for a small timestep δ,
To achieve good precision, δ must be small. In the circuit model, each \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) Trotter step necessarily requires at least one quantum gate to implement. Thus the required circuit depth—and hence the total runtime—is at least T/δ. Contrast this with the runtime if we were able to implement \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) directly in time δ. The total runtime would then be T, which improves on the circuitmodel algorithm by a factor of 1/δ. This is "only” a constant factor improvement, in line with the SolovayKitaev theorem. But this "constant” can be very large; indeed, it diverges to ∞ as the precision of the algorithm increases.
It is unrealistic to assume the hardware can implement \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) for any desired interaction h_{ij} and any time δ. Furthermore, the available interactions are typically limited to at most a handful of specific types, determined by the underlying physics of the device’s qubit and quantum gate implementations. And these interactions cannot be switched on and off arbitrarily fast, placing a limit on the smallest achievable value of δ. There are also experimental challenges associated with implementing gates with small δ with the same fidelities as those with δ ≈ O(1).
A major criticism of analogue computation (classical and quantum) is that it cannot cope with errors and noise. The "N” in NISQ stands for "noisy”; errors and noise will be a significant factor in all foreseeable quantum hardware. But nearterm hardware has few resources to spare even on basic error correction, let alone fault tolerance. Indeed, nearterm hardware may not always have the necessary capabilities. E.g. the intermediate measurements required for active error correction are not possible in all superconducting circuit hardware [ref. ^{19}, Sec. II].
Algorithms that cope well with errors and noise, and still give reasonable results without active error correction or fault tolerance, are thus critical for NISQ applications.
Designing algorithms "one level below” the circuit model can also in some cases reduce the impact of errors and noise during the algorithm. Again, this benefit is particularly acute in Hamiltonian simulation algorithms. If an error occurs on a qubit in a quantum circuit, a twoqubit gate acting on the faulty qubit can spread the error to a second qubit. In the absence of any error correction or fault tolerance, errors can spread to an additional qubit with each twoqubit gate applied, so that after circuit depth n the error can spread to all n qubits.
In the circuit model, each \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) Trotter step requires at least one twoqubit gate. So a single error can be spread throughout the quantum computer after simulating timeevolution for time as short as δn. However, if a twoqubit interaction \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) is implemented directly, one would intuitively expect it to only "spread the error” by a small amount δ for each such timestep. Thus we might expect it to take time O(n) before the error can propagate to all n qubits—a factor of 1/δ improvement. Another way of viewing this is that, in the circuit model, the LiebRobinson velocity^{20} at which effects propagate in the system is always O(1), regardless of what unitary dynamics is being implemented by the overall circuit. In contrast, the Trotterized Hamiltonian evolution has the same LiebRobinson velocity as the dynamics being simulated: O(1/δ) in the same units.
The FermiHubbard model is believed to capture, in a simplified toy model, key aspects of hightemperature superconductors, which are still less well understood theoretically than their lowtemperature brethren. Its Hamiltonian is given by a combination of onsite and hopping terms:
describing electrons with spin σ = ↑ or ↓ hopping between neighbouring sites on a lattice, with an onsite interaction between oppositespin electrons at the same site. The FermiHubbard model serves as a particularly good testbed for NISQ Hamiltonian simulation algorithms for a number of reasons [ref. ^{21}, Sec. IV], beyond the fact that it is a scientifically interesting model in its own right:

(1)
The FermiHubbard model was a famous, wellstudied condensed matter model long before quantum computing was proposed. It is therefore less open to the criticism of being an artificial problem tailored to fit the algorithm.

(2)
It is a fermionic model, which poses particular challenges for simulation on (qubitbased) quantum computers. Most of the proposed practical applications of quantum simulation involve fermionic systems, either in quantum chemistry or materials science. So achieving quantum simulation of fermionic models is an important step on the path to practical quantum computing applications.

(3)
There have been over three decades of research developing evermoresophisticated classical simulations of FermiHubbardmodel physics^{22}. This gives clear benchmarks against which to compare quantum algorithms. And it reduces the likelihood of there being efficient classical algorithms, which have not been discovered because little interest or effort has been devoted to the model.
The stateoftheart quantum circuitmodel algorithm for simulating the time dynamics of the 2D FermiHubbard model on an 8 × 8 lattice requires ≈10^{7} Toffoli gates [ref. ^{15}, Sec. C: Tb. 2]. This includes the overhead for fault tolerance, which is necessary for the algorithm to achieve reasonable precision with the gate fidelities available in current and nearterm hardware. But it additionally incorporates performing phase estimation, which is a significant extra contribution to the gate count. Thus, although this result is indicative of the scale required for standard circuitmodel Hamiltonian simulation, a direct comparison of this result with timedynamics simulation would be unfair.
To establish a fair benchmark, using available Trotter error bounds from the literature^{23} with the best previous choice of fermion encoding in the literature^{24}, we calculate that one could achieve a FermiHubbard timedynamics simulation on a 5 × 5 square lattice, up to time T = 7 and to within 10% accuracy, using 50 qubits and 1,243, 586 standard twoqubit gates. This estimate assumes the effects of decoherence and errors in the circuit can be neglected, which is certainly overoptimistic.
Our results rely on developing more sophisticated techniques for synthesising manybody interactions out of the underlying one and twoqubit interactions available in the quantum hardware (see Results). This gives us access to \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) for more general interactions h_{ij}. We then quantify the type of gains discussed here under two precisely defined error models, which correspond to different assumptions about the hardware. By using the aforementioned techniques to synthesise local Trotter steps, exploiting a recent fermion encoding specifically designed for this type of algorithm^{25}, deriving tighter error bounds on the higherorder Trotter expansions that account for all constant factors, and carefully analysing analytically and numerically the impact and rate of spread of errors in the resulting algorithm, we improve on this by multiple orders of magnitude even in the presence of decoherence. For example, we show that a 5 × 5 FermiHubbard timedynamics simulation up to time T = 7 can be performed to 10% accuracy in what we refer to as a pergate error model with ≈50 qubits and the equivalent of circuit depth 72,308. This is a conservative estimate and based on analytic Trotter error bounds that we derive in this paper. Using numerical extrapolation of Trotter errors, a circuit depth of 3209 can be reached. In the second error model, which we refer to as a pertime error model, we prove rigorously that the same simulation is achievable in a circuitdepthequivalent runtime of 1686; numerical error computations bring this down to 259. In the pertime model, for some parameter regimes we are also able to exploit the inherent partial errordetection properties of local fermionic encodings to enable error mitigation strategies to reduce the resource cost. This brings Hamiltonian simulation, previously beyond reach of current hardware for nontrivial examples, significantly closer to being feasible in the NISQ era.
Results and discussion
Circuit error models
We consider two error models for quantum computation in this work. The first error model assumes that noise occurs at a constant rate per gate, independent of the time it takes to implement that gate. This is the standard error model in quantum computation theory, in which the cost of a computation is proportional to its circuit depth. We refer to this model as the pergate error model. The second error model assumes that noise occurs at a constant rate per unit time. This is the traditional model of errors in physics, where dissipative noise is more commonly modelled by continuoustime master equations, which translates to the pertime error model. In this model, the errors accumulate proportionately to the time the interactions in the system are switched on, thus with the total pulse lengths. We refer to this as the pertime error model We emphasise that these error models are not fundamentally about execution time, but about an error budget required to execute a particular circuit. While it is clear that deeper circuits experience more decoherence, how much each gate contributes to it can be analysed from two different perspectives. The two error models we study correspond to two difference models of how noise scales in quantum hardware.
Which of these more accurately models errors in practice is hardware dependent. For example, in NMR experiments, the pertime model is common^{26,27,28}. The pertime model is not without basis in more recent quantum hardware, too. Recent work has developed and experimentally tested durationscaled twoqubit gates using Qiskit Pulse and IBM devices^{29,30}. In ref. ^{30} the authors experimentally observe braiding of Majorana zero modes using and IBM device and parameterised twoqubit gates. They also find a relationship between relative gate errors and the duration of these parameterised gates, which is further validated in ref. ^{29}. The authors of ref. ^{29} explicitly attribute the reduction in error—seen using these durationscaled gates in place of CNOT gates—to the shorter schedules of the scaled gates relative to the coherence time.
Nonetheless, the standard pergate error model is also very relevant to current quantum hardware hardware. Therefore, throughout this paper we carry out full error analyses of all our algorithms in both of these error models.
Both of these error models are idealisations. Both are reasonable from a theoretical perspective and supported by certain experiments. Analysing both error models allows different algorithm implementations to be compared fairly under different error regimes. In particular, analysing both of these error models gives a more stringent test of new techniques than considering only the "standard error model” of quantum computation, which corresponds to the pergate model.
We show that in both error models, significant gains can be achieved using our new techniques.
In our analysis, for simplicity we treat singlequbit gates as a free resource in both error models. There are three reasons for making this simplification, First, singlequbit gates can typically be implemented with an orderofmagnitude higher fidelity in hardware, so contribute significantly less to the error budget than twoqubit gates. Second, they do not propagate errors to the same extent as twoqubit gates (cf. only costing T gates in studies of faulttolerant quantum computation). Third, any quantum circuit can be decomposed with at most a single layer of singlequbit gates between each layer of twoqubit gates. Thus including singlequbit gates in the pergate error model changes the absolute numbers by a constant factor ≤2 in the worst case. Nor does it significantly affect comparisons between different algorithm designs. This is particularly true of productformula simulation algorithms, where the algorithms are composed of the same layers of gates repeated over and over.
Additionally, there is a benefit to utilising our synthesis techniques regardless of error model. Decomposing the simulation into gates of the form \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}{h}_{ij}\delta }\) using these methods allows us to exploit the underlying errordetection properties of fermionic encodings, as explained in Supplementary Methods and demonstrated in Fig. 2 (see below).
Tables 1 and 2 compare these results, showing how the combination of subcircuit algorithms, recent advances in fermion encodings (VC = VerstraeteCirac encoding^{24}, compact = encoding reported in ref. ^{25}), and tighter Trotter bounds (both analytic and numeric) successively reduce the runtime of the simulation algorithm (\({{{{{{\mathcal{T}}}}}}}_{{{{{{\rm{cost}}}}}}}=\) circuit depth for pergate error model, or sum of pulse lengths for pertime error model).
Synthesis of encoded FermiHubbard Hamiltonian Trotter layers
To simulate fermionic systems on a quantum computer, one must encode the fermionic Fock space into qubits. There are many encodings in the literature^{31} but we confine our analysis to two: the VerstraeteCirac (VC) encoding^{24}, and the compact encoding recently introduced in ref. ^{25}. We have selected these two encodings as they minimise the maximum Pauli weight of the encoded interactions, which is a key factor in the efficiency of Trotterbased algorithms and of our subcircuit techniques: weight4 (VC) and weight3 (compact), respectively. By comparison, the classic JordanWigner transformation^{31} results in a maximum Pauli weight that scales as as \({{{{{\rm{O}}}}}}\left(L\right)\) with the lattice size L; the BravyiKitaev encoding^{32} has interaction terms of weight \(O({{{{\mathrm{log}}}}}\,L)\); and the BravyiKitaev superfast encoding^{32} results in weight8 interactions.
Under the compact encoding, the fermionic operators in Eq. (2) are mapped to operators on qubits arranged on two stacked square grids of qubits (one corresponding to the spin up, and one to the spin down sector, as shown in Supplementary Fig. 1d), augmented by a facecentred ancilla in a checkerboard pattern, with an enumeration explained in Supplementary Fig. 1a. The onsite, horizontal and vertical local terms in the FermiHubbard Hamiltonian Eq. (2) are mapped under this encoding to qubit operators as follows:
where qubit \({f}_{ij}^{\prime}\) is the facecentered ancilla closest to vertex (i, j), and g(i, j) indicates an associated sign choice in the encoding, as explained in ref. ^{25}.
If the VC encoding is used, the fermionic operators in Eq. (2) are mapped to qubits arranged on two stacked square grids of qubits (again with one corresponding to spin up, the other to spin down, as shown in Supplementary Fig. 2, augmented by an ancilla qubit for each data qubit and with an enumeration explained in Supplementary Fig. 2a. In this case the onsite, horizontal and vertical local terms are mapped to
where \(i^{\prime} \) indicates the ancilla qubit associated with qubit i.
In both encodings, we partition the resulting Hamiltonian H—a sum of onsite, horizontal and vertical qubit interaction terms on the augmented square lattice—into M = 5 layers H = H_{1} + H_{2} + H_{3} + H_{4} + H_{5}, as shown in Supplementary Figs. 1 and 2. The Hamiltonians for each layer do not commute with one another. Each layer is a sum of mutuallycommuting local terms acting on disjoint subsets of the qubits. For instance, \({H}_{5}={\sum }_{i}{h}_{{{{{{\rm{onsite}}}}}}}^{(i)}\) is a sum of all the twolocal, nonoverlapping, onsite terms.
The Trotter product formula \({{{{{{\mathcal{P}}}}}}}_{p}(T,\delta )\) comprises local unitaries, corresponding to the local interaction terms that make up the five layers of Hamiltonians that we decomposed the FermiHubbard Hamiltonian into.
In order to implement each step of the product formula as a sequence of gates, we would ideally simply execute all two, three (for the compact encoding) or fourlocal (for the VC encoding) interactions necessary for the time evolution directly within the quantum computer. Yet this is an unrealistic assumption, as the quantum device is more likely to feature a very restricted set of one and twoqubit interactions.
As outlined in the introduction, we assume in our model that arbitrary singlequbit unitaries are available, and that we have access to the continuous family of gates \(\{\exp ({{{{{\rm{i}}}}}}tZ\otimes Z)\}\) for arbitrary values of t. In contrast, the gates we wish to implement all have the form \(\exp ({{{{{\rm{i}}}}}}\delta {Z}^{\otimes k})\) for k = 3 or 4. (Or different products of k Pauli operators, but these are all equivalent up to local unitaries, which we are assuming are available.)
It is well known that a unitary describing the evolution under any klocal Pauli interaction can be straightforwardly decomposed into CNOT gates and singlequbit rotations [ref. ^{18},Sec. 4.7.3]. For instance, we can decompose evolution under a 3local Pauli as
where we then further decompose the remaining 2local evolutions in Eq. (9) using the exact same method as
This effectively corresponds to decomposing \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}\delta {Z}_{1}{Z}_{2}{Z}_{3}}\) into CNOT gates and singlequbit rotations, as \({{{{{{\rm{e}}}}}}}^{\pm {{{{{\rm{i}}}}}}\pi /4{Z}_{i}{Z}_{j}}\) is equivalent to a CNOT gate up to singlequbit rotations. To generate evolution under any klocal Pauli interaction we can simply iterate this procedure, which yields a constant overhead ∝ 2(k − 1) × π/4.
Can we do better? Even optimised variants of SolovayKitaev to decompose multiqubit gates—beyond introducing an additional error—generally yield gate sequences multiple orders of magnitude larger, as e.g. demonstrated in ref. ^{33}. While more recent results conjecture that an arbitrary threequbit gate can be implemented with at most eight O(1) twolocal entangling gates^{34}, this is still worse than the conjugation method for the particular case of a rank one Pauli interaction that we are concerned with.
For small pulse times δ, the existing decompositions are thus inadequate, as they all introduce a gate cost Ω(1) + O(δ). In this paper, we develop a series of analytic pulse sequence identities (see Supplementary Lemmas 7 and 8 in Supplementary Methods, which allow us to decompose the threequbit and fourqubit gates as approximately The approximations in Eqs. (11) and (12) are shown to first order in δ. Exact analytic expressions, which also hold for δ ≥ 1, are derived in Supplementary Methods. The constants in Eq. (12) have been rounded to the third significant figure.
In reality we use the exact versions of these decompositions, which we also note are still exact for δ ≥ 1. The depth5 decomposition in Eq. (12) yields the shortest overall runtime when breaking down higherweight interactions in a recursive fashion, assuming that the remaining threelocal gates are decomposed using an expression similar to Eq. (11). We also carry out numerical studies that indicate that these decompositions are likely to be optimal. (See Supplementary Methods). These circuit decompositions allow us to establish that, for a weightk interaction term, there exists a pulse sequence which implements the evolution operator for time δ with an overhead ∝ δ^{1/(k−1)}, achieved by recursively applying these decompositions. While we have only made reference to interactions of the form Z^{⊗k}, we remark that this is sufficient as we can obtain any other interaction term of the same weight, for example ZXZ, by conjugating Z^{⊗k} by singlequbit rotations, H and SHS^{†} in this example (where H is a Hadamard and S a phase gate).
For the interactions required for our FermiHubbard simulation, the overhead of decomposing shortpulse gates with this analytic decomposition is \(\propto \sqrt{\delta }\) for any weight3 interaction term, and ∝δ^{1/3} for weight4. The asymptotic runtime is thus \({{{{{\rm{O}}}}}}(T{\delta }_{0}^{w})\) for w = −1/2 (compact encoding) or w = − 2/3 (VC encoding). We show the exact scaling for k = 3 and k = 4 in Fig. 1, as compared to the standard conjugation method.
Tighter error bounds for Trotter product formulas
There are by now a number of sophisticated quantum algorithms for Hamiltonian simulation, achieving optimal asymptotic scaling in some or all parameters^{35,36,37}. Recently^{38}, have shown that previous error bounds on Trotter product formulae were overpessimistic. They derived new bounds showing that the older, simpler, productformula algorithms achieve almost the same asymptotic scaling as the more sophisticated algorithms.
For nearterm hardware, achieving good asymptotic scaling is almost irrelevant; what matters is minimising the actual circuit depth for the particular target system being simulated. Similarly, in the NISQ regime we do not have the necessary resources to implement full active error correction and fault tolerance. But we can still consider ways of minimising the output error probability for the specific computation being carried out. Simple productformula algorithms allow good control of error propagation in the absence of active error correction and fault tolerance. Furthermore, combining productformula algorithms with our circuit decompositions allows us to exploit the errordetection properties of fermionic encodings. We can use this to relax the effective noise rates required for accurate simulations, especially if we are willing to allow the simulation to include some degree of simulated natural noise. This is explained further the Supplementary Methods and the results of this technique are shown in Fig. 2.
For these reasons, we choose to implement the timeevolution operator \(U(T):=\exp ({{{{{\rm{i}}}}}}TH)\) by employing Trotter product formulae \(U(T)=:{{{{{{\mathcal{P}}}}}}}_{p}{(\delta )}^{T/\delta }+{{{{{{\mathcal{R}}}}}}}_{p}(T,\delta )\). Here, \({{{{{{\mathcal{R}}}}}}}_{p}\left(T,\delta \right)\) denotes the error term remaining from the approximate decomposition into a product of individual terms, defined directly as \({{{{{{\mathcal{R}}}}}}}_{p}\left(T,\delta \right):=U(T){{{{{{\mathcal{P}}}}}}}_{p}{\left(\delta \right)}^{T/\delta }\). This includes the simple firstorder formula^{13}
as well as higherorder variants^{38,39,40}
for \(k\in {\mathbb{N}}\), where the coefficients are given by \({a}_{k}:=1/\left(4{4}^{1/\left(2k1\right)}\right)\). It is easy to see that, while for higherorder formulas not all pulse times equal δ, they still asymptotically scale as Θ(δ). The product formula \({{{{{{\mathcal{P}}}}}}}_{p}{\left(\delta \right)}^{T/\delta }\) then approximates a time evolution under U(δ)^{T/δ} ≈ U(T), and it describes the sequence of local unitaries to be implemented as a quantum circuit.
Choosing the Trotter step δ small means that corrections for every factor in this formula come in at \({{{{{\rm{O}}}}}}\left({\delta }^{p+1}\right)\) for \(p\in \{1,2k:k\in {\mathbb{N}}\}\). Since we have to perform T/δ many rounds, the overall error scales roughly as \({{{{{\rm{O}}}}}}\left(T{\delta }^{p}\right)\). Yet this rough estimate is insufficient if we need to calculate the largestpossible δ for our Hamiltonian simulation.
The Hamiltonian dynamics remain entirely within one fermion number sector, as H_{FH} commutes with the total fermion number operator. Let Λ denote the number of fermions present in the simulation, such that ∥H_{i}∣_{Λ fermions}∥ ≤ Λ as shown in Supplementary Theorem 23. Let M = 5 denote the number of noncommuting Trotter layers, and set \({\epsilon }_{p}(T,\delta ):=\parallel {{{{{{\mathcal{R}}}}}}}_{p}(T,\delta )\parallel \), and as shorthand ϵ_{p}(δ): = ϵ_{p}(δ, δ), so that ϵ_{p}(T, δ) = T/δ × ϵ_{p}(δ).
To obtain a bound on \({{{{{{\mathcal{P}}}}}}}_{p}\left(\delta \right)\), we apply the variation of constants formula [ref. ^{41},Th, 4.9] to \({{{{{{\mathcal{R}}}}}}}_{p}(\delta )\), with the condition that \({{{{{{\mathcal{P}}}}}}}_{p}\left(0\right)={\mathbb{1}}\), which always holds. As in [ref. ^{38}, sec. 3.2], for δ ≥ 0, we obtain
where the integrand \({R}_{p}\left(\tau \right)\) is defined as
Now, if \({{{{{{\mathcal{P}}}}}}}_{p}\left(\delta \right)\) is accurate up to pth order—meaning that \({{{{{{\mathcal{R}}}}}}}_{p}\left(\delta \right)={{{{{\rm{O}}}}}}\left({\delta }^{p+1}\right)\)—it holds that the integrand \({R}_{p}\left(\delta \right)={{{{{\rm{O}}}}}}\left({\delta }^{p}\right)\). This allows us to restrict its partial derivatives for all 0 ≤ j ≤ p − 1 to \({\partial }_{\tau }^{j}{R}_{p}\left(0\right)=0\). For full details see Supplementary Lemma 13 and [ref. ^{38}, Order Conditions].
Then, following ref. ^{38}, we perform a Taylor expansion of \({R}_{p}\left(\tau \right)\) around τ = 0, simplifying the error bound \({\epsilon }_{p}(\delta )\equiv \parallel {{{{{{\mathcal{R}}}}}}}_{p}(\delta )\parallel \) to
Here we use the aforementioned order condition that for all 0 ≤ j ≤ p − 1 the partial derivatives satisfy \({\partial }_{\tau }^{j}{R}_{p}\left(0\right)=0\), leaving all but the p^{th} or higher remainder terms—\({S}_{p}\left(\tau ,0\right)\)—equal to zero. Thus
where we used the integral representation for the Taylor remainder \({S}_{p}\left(\tau ,0\right)\).
Motivated by this, we look for simple bounds on the pth derivative of the integrand \(\parallel {R}_{p}\left(\tau \right)\parallel \). At this point our work diverges from ref. ^{38} by focusing on obtaining bounds on \(\parallel {R}_{p}\left(\tau \right)\parallel \), which have the tightest constants for NISQera system sizes, but which now are not optimal in system size. (See Supplementary Fig. 6 and Supplementary Lemmas 14 and 19 in Supplementary Methods for details.) We derive the following explicit error bounds (see Supplementary Theorem 15 and Supplementary Corollary 16):
where
and
where
The above expressions hold for generic Trotter formulae. Using Supplementary Lemma 19 we can exploit commutation relations for the specific Hamiltonian at hand (whose structure determines N and n, see Supplementary Methods). This yields the bound (see Supplementary Theorem 20):
where
and
These analytic error bounds are then combined with a TaylorofTaylor method, by which we expand the Taylor coefficient \({R}_{p}^{(p)}\) in Eq. (21) itself in terms of a power series to some higher order q > p, with corresponding series coefficients \({R}_{p}^{(q)}\), and a corresponding remainderofremainder error term ϵ_{p,q+1}. The tightest error expression we obtain is (see Supplementary Corollary 22 in Supplementary Methods)
where the f(p, M, l) are exactly calculated coefficients (using a computer algebra package) that exploit cancellations between the M noncommuting Trotter layers, for a product formula of order p and series expansion order l (given in Supplementary Table 1). The series’ remainder ϵ_{p,q+1} therein is then derived from the analytic bounds in Eq. (26) (see Supplementary Methods for technical details).
Henceforth, we will assume the tightest choice of ϵ_{p}(δ) among all the derived error expressions and choice of p ∈ {1, 2, 4}. In order to guarantee a target error bound ϵ_{p}(T, δ) ≤ ϵ_{target}, we invert these explicitly derived error bounds and obtain a maximum possible Trotter step δ_{0} = δ_{0}(ϵ_{target}).
Benchmarking the subcircuitmodel
How significant is the improvement of the measures set out in previous sections, as benchmarked against stateoftheart results from literature? A first comparison is in terms of exact asymptotic bounds (which we derive in Supplementary Corollaries 25 and 26), in terms of the number of noncommuting Trotter layers M, fermion number Λ, simulation time T and target error ϵ_{target}:
Standard circuit synthesis:
Subcircuit synthesis:
Here we write \({{{{{{\mathcal{T}}}}}}}_{{{{{{\rm{cost}}}}}}}\) for the "runtime” of the quantum circuits—i.e. the sum of pulse times of all gates within the circuit (See Supplementary Definition 3 for a detailed discussion of the cost model we employ).
Beyond asymptotic scaling, and in order to establish a more comprehensive benchmark that takes into account potentially large but hidden constant factors, we employ our tighter Trotter error bounds that account for all constant factors, and concretely target a 5 × 5 FermiHubbard Hamiltonian for overall simulation time T = 7 (which is roughly the LiebRobinson time required for the "causalitycone” to spread across the whole lattice, and for correlations to potentially build up between any pair of sites), in the sector of Λ = 5 fermions, and coupling strengths ∣u∣, ∣v∣ ≤ r = 1 as given in Eq. (2). For this system, we choose the optimal Trotter productformula order p that yields the lowest overall runtime, while still achieving a target error of ϵ_{target} = 0.1.
The results are given in Tables 1 and 2, where we emphasise that in order to maintain a fair comparison, we always account singlequbit gates as a free resource, for the reasons discussed in the Introduction, and twoqubit gates are either accounted at one unit of time per gate in the pergate error model (making the runtime equal the circuit depth), or accounted at their pulse length for the pertime error model.
Our Trotter error bounds yield an orderofmagnitude improvement as compared to [ref. ^{23}, Prop F.4]. And even for existing gate decompositions by conjugation, the recently published lowerweight compact encoding yields a small but significant improvement. The most striking advantage comes from utilising the subcircuit sequence decompositions developed in this paper, in particular in conjunction with the lowerweight compact fermionic encoding.
Overall, the combination of Trotter error bounds, numerics, compact fermion encoding and subcircuitmodel algorithm design, allows us to improve the runtime of the simulation algorithm from 976,710 to 259—an improvement of more than three orders of magnitude over that obtainable using the previous stateoftheart methods, and a further improvement over results in the preexisting literature^{15}.
Subcircuit algorithms on noisy hardware
As ours is a study of quantum simulation on nearterm hardware, we cannot neglect decoherence errors that inevitably occur throughout the simulation. To address this concern, we assume an iid noise model described by the qubit depolarising channel
applied to each individual qubit in the circuit, and after each gate layer in the Trotter product formula, such that the bit, phase, and combined bitphaseflip probability q is proportional to the elapsed time of the preceding layer. While this standard error model is simplistic, it is a surprisingly good match to the errors seen in some hardware^{6}.
Within this setting, a simple analytic decoherence error bound can readily be derived (see Supplementary Methods), by calculating the probability that zero errors appear throughout the circuit. If V denotes the volume of the circuit (defined as \({{{{{{\mathcal{T}}}}}}}_{{{{{{\rm{cost}}}}}}}\times {L}^{2}\)), then the depolarising noise parameter \(q \, < \, 1{(1{\epsilon }_{{{{{{\rm{target}}}}}}})}^{1/V}\)—i.e. it needs to shrink exponentially quickly with the circuit’s volume. We emphasise that this is likely a crude overestimate. As briefly discussed at the start, one of the major advantages of subcircuit circuits is that, under a shortpulse gate, an error is only weakly propagated due to the reduced LiebRobinson velocity (discussed further in ref. ^{42}).
Yet irrespective of this overestimate, can we derive a tighter error bound by other means? In ref. ^{42}, the authors analyse how noise on the physical qubits translates to errors in the fermionic code space. To first order and in the compact encoding, all of {X, Y, Z} errors on the face, and {X, Y} on the vertex qubits can be detected. Z errors on the vertex qubits result in an undetectable error, as evident from the form of h_{onsite} from Supplementary Equation (140). It is shown in [ref. ^{42}, Sec. 3.2] that this Z error corresponds to fermionic phase noise in the simulated FermiHubbard model.
It is therefore a natural extension to the notion of simulation to allow for some errors to occur, if they correspond to physical noise in the fermionic space. And indeed, as discussed more extensively in [ref. ^{42}, Sec. 2.4], phase noise is a natural setting for many fermionic condensed matter systems coupled to a phonon bath^{43,44,45,46,47,48,49} and [ref. ^{50}, Ch. 6.1&eq. 6.17].
How can we exploit the encoding’s error mapping properties? Under the assumption that X, Y and Z errors occur uniformly across all qubits, as assumed in Eq. (33), each Pauli error occurs with probability q/3. We further assume that we can measure all stabilizers (including a global parity operator) once at the end of the entire circuit, which can be done by dovetailing a negligible depth 4 circuit to the end of our simulation (see Supplementary Methods for more details). We then numerically simulate a stochastic noise model for the circuit derived from aforementioned Trotter formula for a specific target error ϵ_{target}, for a FermiHubbard Hamiltonian on an L × L lattice for L ∈ {3, 5, 10}.
Whenever an error occurs, we keep track of the syndrome violations they induce (including potential cancellations that happen with previous syndromes), using results from ref. ^{42} on how Pauli errors translate to error syndromes with respect to the fermion encoding’s stabilizers (summarised in Supplementary Table 2). We then bin the resulting circuit runs into the following categories:

(1)
Detectable error: at least one syndrome remains triggered, even though some may have canceled throughout the simulation,

(2)
Undetectable phase noise: no syndrome was ever violated, and the only errors are Z errors on the vertex qubits which map to fermionic phase noise, and

(3)
Undetectable nonphase noise: syndromes were at some point violated, but they all canceled.

(4)
Errors not happening in between Trotter layers: naturally, not all errors happen in between Trotter layers, so this category encompasses all those cases where errors happen in between gates in the gate decomposition.
This categorisation allows us to calculate the maximum depolarising noise parameter q to be able to run a simulation for time \(T=\lfloor \sqrt{2}L\rfloor \) with target Trotter error ϵ_{t} ≤ ϵ_{target} ∈ {1%, 5%, 10%}, where we allow the resulting undetectable nonphase noise and the errors not happening in between Trotter layers errors to also saturate this error bound, i.e. ϵ_{s} ≤ ϵ_{target}. The overall error is thus guaranteed to stay below a propagated error probability of \({({\epsilon }_{{{{{{\rm{t}}}}}}}^{2}+{\epsilon }_{s}^{2})}^{1/2}\in \{1.5 \% ,7.1 \% ,15 \% \}\), respectively.
In order to achieve these decoherence error bounds, one needs to postselect "good” runs and discard ones where errors have occurred, as determined from the single final measurement of all stabilizers of the compact encoding. The required overhead due to the postselected runs is mild, and shown in Supplementary Fig. 20.
We plot the resulting simulation cost vs. target simulation time in Fig. 2 and Supplementary Figs. 16–19 where we colour the graphs according to the depolarising noise rate required to achieve the target error bound. For instance, in the tightest pertime error model (bottom right plot in Fig. 2), a depolarising noise parameter q = 10^{−5} allows simulating a 5 × 5 FH Hamiltonian for time T ≈ 5, while satisfying a 15% error bound, the required circuitdepthequivalent is \({{{{{{\mathcal{T}}}}}}}_{{{{{{\rm{cost}}}}}}}\approx 140\)—and for time T ≈ 2.5 for a 7.1% error bound, for \({{{{{{\mathcal{T}}}}}}}_{{{{{{\rm{cost}}}}}}}\approx 70\).
In this work, we have derived a method for designing quantum algorithms "one level below” the circuit model, by designing analytic subcircuit identities to decompose the algorithm into. As a concrete example, we applied these techniques to the task of simulating timedynamics of the spin FermiHubbard Hamiltonian on a square lattice. Together with Trotter product formulae error bounds applied to the recent compact fermionic encoding, we estimate these techniques provide a three orders of magnitude reduction in circuitdepthequivalent. The authors of ref. ^{51} have recently extended their work on error bounds in ref. ^{51}, beyond their results in ref. ^{38}. We have not yet incorporated their new bounds into our analysis, and this may give further improvements over our analytic error bounds.
Naturally, any real world implementation on actual quantum hardware will allow and require further optimisations; for instance, all errors displayed within this paper are in terms of operator norm, which indicates the worstcase error deviation for any simulation. However, when simulating timedynamics starting from a specific initial configuration and a distinct final measurement setup, a lower error rate results. We have accounted for this in a crude way, by analysing simulation of the FermiHubbard model dynamics with initial states of bounded fermion number. But the error bounds—even the numerical ones—are certainly pessimistic for any specific computation. Furthermore, while we already utilise numerical simulations of Trotter errors, more sophisticated techniques such as Richardson extrapolation for varying Trotter step sizes might show promise in improving our findings further.
It is conceivable that other algorithms that require small unitary rotations will similarly benefit from designing the algorithms “one level below” the circuit model. Standard circuit decompositions of many interesting quantum algorithms will remain unfeasible on real hardware for some time to come. Whereas our subcircuitmodel algorithms, with their shorter overall runtime requirements and lower error propagation even in the absence of error correction, potentially bring these algorithms and applications within reach of nearterm NISQ hardware.
Code availability
The code to support the findings in this work is available upon request from the authors.
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Acknowledgements
We thank Joel Klassen for providing the proof of Supplementary Theorem 23, and for many useful discussions. Laura Clinton is part funded by EPSRC.
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Clinton, L., Bausch, J. & Cubitt, T. Hamiltonian simulation algorithms for nearterm quantum hardware. Nat Commun 12, 4989 (2021). https://doi.org/10.1038/s41467021251960
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DOI: https://doi.org/10.1038/s41467021251960
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