Resource-optimized fault-tolerant simulation of the Fermi-Hubbard model and high-temperature superconductor models

Abstract

Exploring low-cost applications is paramount to creating value in early fault-tolerant quantum computers. Here, we optimize both gate and qubit counts of recent algorithms for simulating the Fermi-Hubbard model. We further devise and compile algorithms to simulate established models of cuprate and pnictide high-temperature superconductors, which include beyond-nearest-neighbor hopping terms and multi-orbital interactions that are absent in the Fermi-Hubbard model. We show that simulations of these more realistic models of high-temperature superconductors require only an order of magnitude or so more Toffoli gates than a simulation of the Fermi-Hubbard model. Furthermore, we find plenty classically difficult instances with Toffoli and qubit counts that are far lower than commonly considered quantum phase estimation circuits for electronic structure problems in quantum chemistry. We believe our results pave the way towards studying high-temperature superconductors on early fault-tolerant quantum computers.

Introduction

The Fermi-Hubbard (FH) model¹ is a longstanding model of interacting electronic systems, which despite its ostensibly simple definition, contains rich physics and is notoriously difficult to simulate classically². The FH model has become a popular subject of fault-tolerant quantum simulation and is widely expected to be one of the earliest physical models to be simulated on a fault-tolerant quantum computer (FTQC) due to its low computational resource requirements.

In particular, energy estimation algorithms for the FH model on a L × L square lattice have undergone various optimizations and resource estimations under a fault-tolerant cost model^{3, 4, 5, 6, 7–8}. In⁴, Babbush et al. constructed and compiled such an algorithm based on qubitization⁹ down to Clifford + T gates, albeit neglecting the sub-leading order contribution from rotation synthesis. In ref. ⁵, Kivlichan et al. devised and analyzed an algorithm based on second-order Trotter formulas¹⁰. They showed that under an extensive error model, i.e., where the error is O(N) where N is the system size (They argued that the extensive error model is acceptable in some condensed matter simulations, when, for example, the target observables are robust against a single quasi-particle excitation (see ref. ⁴ for a more detailed discussion).), the Trotter algorithm can be performed with O(1) T complexity for practically relevant system sizes, compared to the O(N) scaling of the algorithm in ref. ⁴. Under the more lenient, extensive error model, simulations can be performed using orders of magnitude fewer gates, which is favorable to the early FTQC era. More recently, Campbell proposed a new second-order Trotter algorithm with significantly reduced Toffoli count⁷.

In this work, we optimize and compare the computational resource requirements of the qubitization algorithm⁴ and Trotter algorithm⁷, under the same extensive error (Although we focus on the extensive error model, our optimizations apply to the intensive error model as well.) in refs. ^7,11, i.e., 0.51%L². We adopt Toffoli (Here we assume that 2 T gates can be performed using 1 Toffoli gate with catalyst-assisted circuits from¹².) and logical-qubit count as our cost metrics, which is standard practice in the fault-tolerant algorithm literature, e.g.,⁷. Despite the expected poorer asymptotic scaling of the qubitization algorithm in this error regime, we still consider it because constant factors are important for small but classically difficult system sizes.

An often-cited motivation behind algorithms to simulate the FH model is to study high-temperature superconductors. It was stated in ref. ⁵ that the FH model was a candidate model for cuprate high-temperature superconductors, based on various classical simulations in ref. ². In ref. ⁷, it was stated that the FH model might elucidate the mechanisms behind high-temperature superconductivity. However, more recent numerical evidence^13,14 suggests that in the parameter regime most relevant to cuprate high-temperature superconductors, the FH model does not exhibit superconductivity.

We take the perspective that, to reliably study high-temperature superconductors on a quantum computer, we need to simulate well-established models. We take a step in this direction by designing and compiling Trotter and qubitization algorithms to simulate a single-orbital model for cuprate superconductors^{15, 16, 17, 18, 19, 20–21} and a two-orbital model for iron-pnictide superconductors^22,23. These models are more complex than the FH model, as they contain beyond-nearest-neighbor hopping and multi-orbital interactions, and as such, are harder to classically simulate. To our knowledge, there does not exist any explicit quantum algorithm with resource estimation for simulating such models. This work fills this gap and demonstrates that simulations of the considered more realistic models of high-temperature superconductors only require an increase in Toffoli counts by an order of magnitude or so, when compared to a simulation of the FH model. Furthermore, they can be simulated with fewer computational resources than quantum phase estimation of the electronic structure of many commonly considered molecules^{24, 25, 26, 27–28}.

Results

Trotter schemes for beyond-nearest-neighbor and multi-orbital interactions

Models of high-temperature superconductors, such as the cuprate and pnictide models considered here, often contain interactions beyond nearest neighbors and between distinct orbitals. Devising Trotter schemes with low Trotter errors and efficient circuit implementations is essential to low-cost simulations of such models. To this end, we design Trotter schemes, for which the Trotter error bounds have constants similarly in magnitude to the constants in the FH Trotter error bounds^7,29.

In our schemes, the circuit implementation of the long-range hopping terms shares a similar structure to that of the nn hopping in ref. ⁷, thereby maximizing the use of HWP. There are three and four types of hopping terms, classified by the magnitudes of their coupling coefficients, in the cuprate and pnictide models, respectively. We implement each type in at most 4 steps (labelled by l in (8)). Upon applications of appropriate fermionic-swap circuits, consisting of only Clifford gates¹¹, each segment can be localized to n n hopping terms acting on either non-overlapping (i) plaquettes, as depicted in Fig. 1a, or (ii) links. For example, the second-nn hopping terms in Fig. 1b will be mapped to case (ii) if we swap to every even site on every even row, assuming a zero-based numbering, with the site directly above. Case (i) can be implemented using the aforementioned circuit from ref. ⁷. The circuit implementation of case (ii) is even simpler. The hopping term on each link is of the form e^iθ(XX+YY), which can be diagonalized to a layer of two same-angle R_z gates using Clifford gates^5,30. As such, case (ii) can be implemented using HWP, up to Clifford gates. The on-site Coulomb term in the cuprate model and the intra-orbital Coulomb term in the pnictide model are implemented the same way as the Coulomb term in the Fermi-Hubbard model; the implementation of the inter-orbital Coulomb term in the pnictide model is almost the same, except fermionic swaps are used to localize them to two-body ZZ operators, which are then implemented using Clifford gates and HWP. We defer further details of the Trotter schemes for the three models to Supplementary Note 2.

Fig. 1 Visualization of Trotter schemes. [Images not available. See PDF.]

a Half of the nn hopping terms in the FH model and cuprate model depicted by the red plaquettes. The other half is obtained by a lattice translation of e_x + e_y. The halves have evolved separately. b A quarter of the second-nn hopping terms in the cuprate model depicted by the red crosses. The remaining three quarters are obtained by lattice translations of e_x, e_y, and e_x + e_y. Each quarter is evolved separately. c Half of the third-nn hopping terms in the cuprate model are depicted by both the solid red and dashed lines. The remaining half is obtained by a lattice translation of 2(e_x + e_y). The halves have evolved separately.

Space-time trade-off in Hamming-weight phasing

In the standard, baseline HWP^7,31,32, the Hamming weight of the to-be-rotated M-qubit state is first computed into an ancilla register of size $\lfloor {\log }_2\left(M\right)\rfloor +1$ . Then, $\lfloor {\log }_2\left(M\right)\rfloor +1R_z$ rotations ${⨂}_{i=0}^{\lfloor {\log }_2\left(M\right)\rfloor }{R}_z\left(2^i\theta \right)$ are applied to the ancilla register before the Hamming weight is uncomputed. It was shown in ref. ³³ that the Hamming weight of a binary register can be computed using M − w(M) half- and full-adders, where w(M) is the number of 1s in the binary representation of M. Briefly, starting from the lowest bit of the binary register, we perform a chain of summations, using full adders as much as possible; each sum bit becomes an input to the next adder until the last adder, and the carry bits become inputs to the next chain of summations. This carries on recursively until there are no more bits to add. A half- or full-adder can be implemented, up to Clifford gates, with a Toffoli gate, and can be uncomputed via a measurement-feed-forward Clifford circuit³¹. (See Supplementary Fig. 2 for the half- and full-adder circuits, and an example of an 8-bit Hamming-weight computation circuit.) This means that the computation and un-computation of the Hamming-weight function cost M − w(M) Toffoli gates.

We introduce a gate-optimized version of HWP, which we call catalyzed HWP. In catalyzed HWP, ${⨂}_{i=0}^{\lfloor {\log }_2\left(M\right)\rfloor }{R}_z\left(2^i\theta \right)$ is implemented via a generalized phase gradient operation (see equation 168 in ref. ³⁴; also shown in Supplementary Fig. 3). Briefly, the operation applies an adder-like circuit, consisting of $\lfloor {\log }_2\left(M\right)\rfloor +1$ Toffoli gates and a single R_z gate, to a catalyst state of the form ${⨂}_{i=0}^{\lfloor {\log }_2\left(M\right)\rfloor }{R}_z\left(2^i\theta \right)\mid +...+\left(\right.⟩$ and the to-be-rotated target state. Since the catalyst state comes out of the operation unscathed and is thus reusable, its preparation costs are amortized over the course of a simulation. Note that in general, multiple catalyst states are needed to simulate H because each H_l in (8) is associated with a different θ-value and different θ-values require different catalyst states. To summarize, an application of baseline HWP requires M−w(M) Toffoli gates and $\lfloor {\log }_2\left(M\right)\rfloor +1R_z$ rotations, whereas an application of catalyzed HWP requires $M+\lfloor {\log }_2\left(M\right)\rfloor -w\left(M\right)+1$ Toffoli gates and only one R_z, neglecting, for the moment, the synthesis cost of the catalyst state, which we will take into consideration in our resource estimation, and using only logarithmically more ancilla qubits. This trade-off is worthwhile because Toffoli gates are cheaper than R_z gates on a FTQC^12,35.

In the qubit-limited, early FTQC era, it is desirable to limit the number of ancilla qubits. In both baseline and catalyzed HWP, we can perform a layer of same-angle R_z rotations in batches, instead of all at once, which reduces the ancilla-qubit count but sacrifices some gate efficiency. In⁷, the number of same-angle R_z rotations applied per batch was limited to 0.5L² so that ~0.5L² ancilla qubits would be required. We apply baseline and catalyzed HWP, without batching and in batches of 0.5L² rotations, to implement the same-angle R_z gates in the simulations of the FH, cuprate, and pnictide models, and plot their quantum resource estimates in Fig. 2. For the FH model, we set u = 8 and t = 1, which was identified in ref. ² as the parameter setting with the largest uncertainty in classical simulations. For the cuprate model, we set $u=8,t=1,t^{\prime }=0.3$ and t^″ = 0.2, in line with parameter choices for cuprate simulations in the literature^{17, 18, 19, 20–21}. For the pnictide model, we set t₁ = 1, t₂ = 1.3, t₃ = 0.85, t₄ = 0.85, as per²², and u = 8, motivated by the classical hardness of FH simulations at u/t = 8. We refer readers to Supplementary Note 2 for more details on the resource estimation pipeline.

Fig. 2 Toffoli counts for Trotter and qubitization simulations. [Images not available. See PDF.]

Toffoli counts for a Fermi-Hubbard simulations with u = 8 and t = 1, b cuprate simulations with $u=8,t=1,t^{\prime }=0.3$ and t^″ = 0.2, and c pnictide simulations with u = 8, t₁ = 1, t₂ = 1.3, t₃ = 0.85, and t₄ = 0.85. The explicit Toffoli counts in these plots can be found in Supplementary Note 1 and 2. The logical qubit counts of these simulations are shown in Fig. 3. Notably, compared to the FH model, the cuprate and pnictide models require only about one order and less than two orders of magnitude more Toffoli gates to simulate. Note that the resource estimation shown here is carried out for an extensive target error of 0.51%L².

We compare the four different Trotter implementations and highlight the key findings:

• Our Toffoli count estimates for the FH, cuprate, and pnictide simulations plateau around 8 × 10⁵, 5 × 10⁶, and 3 × 10⁷, which are in line with the O(1) T complexity described in ref. ⁵. These numbers are orders of magnitude smaller than typical resource estimates for QPE circuits for moderately sized electronic structure problems^{24, 25, 26, 27–28}, which often require ≳10⁹ Toffoli gates and ≳10³ logical qubits, thereby enriching the early FTQC application landscape. The sub-million Toffoli counts for FH simulations, especially when the system size is addressable with <10³ logical qubits, make them particularly attractive and suitable to become one of the earliest scientific application of FTQCs.

• The implementations based on catalyzed HWP, batched or not, are always more gate-efficient than those based on baseline HWP. In particular, for the smallest classically difficult square system size, i.e., 8 × 8^2,36, a reduction of >20% in Toffoli count is achieved for an FH simulation. Note that the estimates for (batched,) baseline HWP in Fig. 2a are obtained from our re-compilation of the algorithm in ref. ⁷. The gate reductions become smaller at larger system sizes because of the lenient, extensive error model; the improvements in gate efficiency will be more pronounced for applications with smaller error tolerance.

• Simulations using batched, catalyzed HWP can achieve roughly the same or even better gate-efficiency than those using baseline HWP, while employing only a fraction—roughly 1/2, 1/4, and 1/8 for FH, cuprate, and pnictide simulations, respectively—of the ancilla qubits required for baseline HWP.

Qubitization vs. Trotter comparison

We compile qubitization algorithms to simulate the FH, cuprate, and pnictide models, and perform resource estimation for them. Even though their Toffoli complexity, with respect to the system size, is worse than their Trotter counterparts, we want to find out whether there is a range of system sizes where their Toffoli counts are better than or close to their Trotter counterparts. We optimize the qubitization algorithms by applying the same chemical potential shift, i.e., n ↦ n − 1/2, from ref. ⁷, which we have used in our Trotter algorithms. This shift reduces the number of Pauli terms in the Hamiltonian after Jordan-Wigner transformtion; thus, it reduces the induced 1-norm λ and in turn, the Toffoli counts of qubitization algorithms, which we plot in Fig. 2. However, the Toffoli complexity still scales as L², with sub-leading additive costs that scale logarithmically in L, as does the algorithm in ref. ⁴. We find the crossover points in L the linear dimension, where qubitization becomes more expensive in Toffoli count than all Trotter implementations, to be L = 8, 16, and 14 for FH, cuprate, and pnictide simulations, respectively. It is important to point out that qubitization algorithms have a lower ancilla count than Trotter algorithms— $O\left(\log \left(L\right)\right)$ vs. O(L²)—due to the spatial overhead of HWP, as shown in Fig. 3. A potential consequence is that in the qubit-limited, early FTQC era, qubitization simulations of smaller system sizes might still be favorable despite their worse Toffoli count, especially as magic states become cheaper^37,38.

Fig. 3 Logical qubit counts for the same simulations, of which the Toffoli counts are shown in Fig. 2. [Images not available. See PDF.]

The logical qubit counts for simulating the Fermi-Hubbard, cuprate, and pnictide models are shown in a, b, and c, respectively. The numerical values used in these plots can be found in Supplementary Note 1 and Supplementary Note 2. We plot the solid curves as a guide for the eye.

Discussion

In this work, we devise resource-optimized energy estimation algorithms for the Fermi-Hubbard model, a single-orbital model of cuprate superconductors, and a two-orbital model of iron pnictide superconductors. Our algorithms are based on the second-order Trotter formula and qubitization. We introduce a technique called catalyzed Hamming-weight phasing, which is particularly effective at reducing the rotation synthesis costs of our Trotter algorithms. Furthermore, comparing our Trotter and qubitization algorithms, we locate precisely the system sizes where qubitization could be more advantageous, despite its worse gate complexity under the extensive error model. Our resource estimation indicates the simulations of the considered cuprate and pnictide models are mildly costlier than the FH model by one order of magnitude or so in terms of Toffoli counts.

When paired with a preparation protocol for an initial state that has a good overlap with the ground state, which is beyond the scope of this work, our algorithms can be used to perform ground state energy estimation (GSEE)^3,6,39. Typically, the overall cost of GSEE is O(1/γ^b) times the QPE cost, where γ is the overlap and b depends on the chosen method. Research for initial state preparation has been explored in, e.g.,⁶, which applies an adiabatic algorithm to improve matrix-product-state (MPS) estimates in the context of the FH model, and more recently in ref. ^40,41, which propose more optimized algorithms based on MPS. The resource estimates from refs. ^40,41 suggest that the costs of preparing high-quality initial states are similar to or less than the QPE costs, in the context of electronic structure GSEE; whether similar conclusions hold for fermionic lattice simulations is worth investigating by future work. In particular, MPS simulations of the two-dimensional Hubbard model³⁶ could require higher bond dimensions than those of the electronic structure of oft considered molecules, e.g., FeMoco. Since the algorithms in ref. ^40,41 scale with bond dimension, the state preparation cost of Hubbard models could exceed that of FeMoco, which would be more expensive than the QPE cost of Hubbard models. This could potentially be alleviated by considering alternative tensor-network states such as fermionic projected entangled pair states (PEPS), which can achieve similar accuracies as MPS at a much lower bond dimension³⁶. Alternatively, one could consider Lindbladian-based ground-state preparation methods^42,43. For such methods to be efficient, one needs to show that the simulated system’s mixing time scales polynomially with the system size. It has been proven for certain parameter regimes, the Hubbard model exhibits polynomial mixing time⁴³; however, crucially, for the interesting, intermediate U/t-regime, only numerical evidences for fast mixing times exist for small lattices⁴³. In ref. ⁴⁴, one could find positive numerical results for larger lattices, though not of the Hubbard model. In the future, the practicality of any initial state preparation method—whether it be tensor-network or Lindbladian method—will need to be evaluated via rigorous resource estimation, just as the QPE part has been over the years.

Comparisons between classical and quantum GSEE of the FH model are of interest from the quantum—as well as the wider—community; see refs. ^8,45 for quantitative comparisons. The conclusions drawn from any comparison will necessarily depend on the state of both classical and quantum simulation, which are constantly improving, as evidenced by this work. As it currently stands, when considering the FH model on square lattices, 8 × 8 is the largest system size that can be reliably simulated using the state-of-the-art classical algorithms based on MPS^2,36. Simulations of more realistic and complex Hubbard-like models, with beyond-nearest-neighbor hopping and multi-orbital interactions, are likely limited to smaller lattices, although more concrete conclusions are better drawn after more rigorous studies in the future. Since the works of^8,45 and the completion of this work, recent advances in classical tensor-network simulations based on PEPS have been reported in ref. ³⁶; the results therein show ground state energy estimates that are similar but not always better than MPS results at small system sizes and that are, to some extent, in qualitative agreement with theory at system sizes beyond the reach of MPS simulations. We expect larger system sizes to be within the reach of classical tensor-network simulations in the future, and that such developments in classical tensor-network simulations will lead to improvements in quantum initial state preparation protocols, just as classical MPS simulations did. We remark that there could potentially be scientific value in performing quantum simulations even at system sizes that are reachable by classical tensor-network simulations, because the latter are heuristic and tensor-network, e.g., MPS and PEPS, estimates could be improved with more rigorous performance guarantees on a quantum computer using, e.g., algorithms in ref. ⁶. We believe any further nuanced classical-versus-quantum comparisons will continue to garner interest and are best addressed outside of this work.

We list several other interesting directions to be explored in the future below:

• There are other problems beyond GSEE, which could play to the strengths of quantum computers. For example, the estimation of dynamical observables, e.g., dynamical correlations³, which involves time-evolution of a quantum state that can be performed efficiently, using, e.g., our Trotter and qubitization circuits, on a quantum computer but is, to our knowledge, exponentially expensive on a classical computer. Gleaning ground-state observables, other than the energy, are relevant in physics, but can be more expensive than GSEE; see ref. ⁴⁶ for an example in the context of molecular simulations. Developing concrete quantum algorithms to estimate these observables is paramount to expanding the utility of quantum computers.

• We have assumed worst-case Trotter error bounds in our work. However, it is worth investigating the cost reductions made possible when average-case empirical Trotter errors, which are often smaller than worst-case bounds²⁹, are assumed.

• Hubbard-like lattice models are low-energy effective models of real-world strongly-correlated materials¹; they can be derived by keeping very few salient microscopic interactions to make theoretical and numerical studies tractable. The more microscopic interactions a model includes, the more realistic the model is, but it becomes costlier to simulate. It could be interesting to better understand this trade-off by obtaining resource estimates for quantum simulations of cuprate^16,20 and pnictide models⁴⁷ that contain interactions between more orbitals than the models considered in this work.

• Construct simulations of Hubbard-like models interacting with a classical field to probe photoexcitation effects⁴⁸, using time-dependent simulation algorithms⁴⁹.

• Perform a more detailed fault-tolerant cost analysis under computational models that incorporate the cost of Clifford gates, e.g., the active volume model⁵⁰; costing Clifford gates will become more important as magic states become cheaper^37,38.

Methods

Target models

Six decades from its inception¹, the FH model remains intensely studied because it contains very few physical degrees of freedom, yet it captures salient features of strongly correlated electronic systems. Despite the attention it garners, it has been exactly solved only in one dimension⁵¹ and infinite dimensions⁵². In two and three dimensions, which are more relevant to realistic materials, exact solutions remain elusive, and numerical solutions are intractable for even moderate system sizes. Here we focus on the two-dimensional case.

We consider a fermionic Hamiltonian H consisting of a hopping part H_h and on-site Coulomb contribution H_c. H_h is given by

1

$$H_h=\sum _{\begin{array}{c}\hfill \left(\mathit{i},\mathit{j}\right)\in {\mathcal{S}}_1,\hfill \\ \hfill \left(f,f^{\prime }\right)\in {\mathcal{S}}_2\hfill \end{array}}{T}_{\left(\mathit{i},f\right),\left(\mathit{j},f^{\prime }\right)}{a}_{\mathit{i},f}^{†}{a}_{\mathit{j},f^{\prime }},$$ where i, j label lattice sites and $f,f^{\prime }$ label internal degrees of freedom, e.g., spins and orbitals, $T_{\left(\mathit{i},f\right),\left(\mathit{j},f^{\prime }\right)}$ is a Hermitian matrix that stores the coupling coefficients, a^(†) is the annihilation (creation) operator, and the content of sets ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ depend on the model considered. H_c is given by

2

$$H_c=\sum _{\begin{array}{c}\hfill \mathit{i}\in \Omega ,\hfill \\ \hfill \left(f,f^{\prime }\right)\in {\mathcal{S}}_3\hfill \end{array}}{U}_{f,f^{\prime }}{n}_{\mathit{i},f}{n}_{\mathit{i},f^{\prime }},$$ where n = a^†a is the number operator, Ω is the set of all lattice sites, the matrix $U_{f,f^{\prime }}$ stores the on-site coefficients, and ${\mathcal{S}}_3$ is a model-dependent set. We consider an L × L lattice with an even L and periodic boundary conditions such that $\mathrm{\Omega }={\left({\mathbb{Z}}_L\right)}^2={\left\{0,1,...,L-1\right\}}^2$ . Hereafter, we denote the unit vectors (1, 0), (0, 1) ∈ Ω as e_x, e_y, respectively.

For the FH model, ${\mathcal{S}}_1$ contains all nearest-neighbor (nn) pairs, f and $f^{\prime }$ label the spin, ${\mathcal{S}}_2=\left\{\left(\uparrow ,\uparrow \right),\left(\downarrow ,\downarrow \right)\right\}$ , and ${\mathcal{S}}_3=\left\{\left(\uparrow ,\downarrow \right),\left(\downarrow ,\uparrow \right)\right\}$ . The non-zero elements of $T_{\left(\mathit{i},f\right),\left(\mathit{j},f^{\prime }\right)}$ and $U_{f,f^{\prime }}$ are t and u, respectively. For the single-orbital cuprate model^{15, 16, 17, 18, 19, 20–21}, ${\mathcal{S}}_1$ additionally includes second- and third-nn, and $T_{\left(\mathit{i},f\right),\left(\mathit{j},f^{\prime }\right)}$ has three types of distinctly valued, non-zero elements t, $t^{\prime }$ , and t^″, which are the coupling between first-, second-, and third-nearest neighbors, respectively.

For the two-orbital pnictide model^22,23, ${\mathcal{S}}_1$ contains first- and second-nn, f = (σ, d), where σ denotes the spin and d ∈ {x, y} labels the orbital, and ${\mathcal{S}}_2=\left\{\left(\left(\sigma ,d\right),\left(\sigma ,d^{\prime }\right)\right)\mid \sigma \in \left\{\uparrow ,\downarrow \right\},\left(d,d^{\prime }\right)\in \left\{\left(x,x\right),\left(y,y\right),\left(x,y\right),\left(y,x\right)\right\}\right\}$ . $T_{\left(\mathit{i},f\right),\left(\mathit{j},f^{\prime }\right)}$ has five types of distinctly valued, non-zero elements: (i) t₁ couples nearest neighbors in the e_y-direction with orbitals $\left(d,d^{\prime }\right)=\left(x,x\right)$ and those in the e_x-direction with $\left(d,d^{\prime }\right)=\left(y,y\right)$ , (ii) t₂ couples nearest neighbors in the e_x-direction with $\left(d,d^{\prime }\right)=\left(x,x\right)$ and those in the e_y-direction with $\left(d,d^{\prime }\right)=\left(y,y\right)$ , (iii) t₃ couples second-nearest neighbors with the same orbital, (iv) t₄ and (v) − t₄ couple second-nearest neighbors separated by e_x + e_y and e_x − e_y, respectively, with different orbitals. $U_{f,f^{\prime }}$ contains two distinctly valued, non-zero elements: (i) the intra-orbital Coulomb strength u for $\left(f,f^{\prime }\right)\in \left\{\left(\left(\uparrow ,d\right),\left(\downarrow ,d\right)\right)\mid d\in \left\{x,y\right\}\right\}$ and (ii) inter-orbital Coulomb strength v for $\left(f,f^{\prime }\right)\in \left\{\left(\left(\sigma ,x\right),\left({\sigma }^{\prime },y\right)\right)\mid \sigma ,{\sigma }^{\prime }\in \left\{\uparrow ,\downarrow \right\}\right\}$ . Here we do not consider other weaker types of on-site interaction, such as Hund’s rule coupling and pair hopping, that are over an order of magnitude weaker than Coulomb interactions^22,23.

Related work

Energy estimation algorithms for the FH model have been studied extensively^{3, 4, 5, 6, 7–8}. Here we briefly review the state-of-the-art qubitization⁴ and Trotter algorithm⁷, which form the basis of our algorithms.

Qubitization⁹ assumes an input model, where the target Hamiltonian H is expressed as a linear combination of unitaries (LCU), i.e.,

3

$$H=\sum _{i=1}^M{c}_i{P}_i,c_i\in \mathbb{R},c_i\ge 0,$$ where P_i’s are unitary and self-inverse, e.g., Pauli operators. Then, H is encoded using the following oracles:

4

$$\mathrm{PREP}\mid 0\left(\right.⟩=\sum _{i=1}^M\sqrt{\frac{c_i}{\lambda }}\mid i\left(\right.⟩,\lambda =\sum _{i=1}^M{c}_i,$$

5

$$\mathrm{SEL}=\sum _{i=1}^M\mid i\left(\right.⟩\left.⟨\right)i\mid \otimes P_i,$$

6

$$\Rightarrow \left(\left.⟨\right)0\mid \otimes I\right){\mathrm{PREP}}^{†}\cdot \mathrm{SEL}\cdot \mathrm{PREP}\left(\mid 0\left(\right.⟩\otimes I\right)=\frac{H}{\lambda },$$ where I is the identity acting on the same register as P_i. As such, PREP^† ⋅ SEL ⋅ PREP is a block-encoding of H⁹.

In ref. ⁴, the FH Hamiltonian is transformed into a LCU, with P_i’s being Pauli operators, using the Jordan-Wigner (JW) transformation⁵³. Then, an energy estimation algorithm is constructed by applying quantum phase estimation (QPE) to a walk operator $\mathcal{W}$ , which shares the same eigenvalues as $e^{i\arccos \left(H/\lambda \right)}$ and $e^{-i\arccos \left(H/\lambda \right)}$ , and can be constructed from a block-encoding and a multiply controlled Z gate^4,39. Further shown in ref. ⁴ is the computational resource estimation of the algorithm in terms of T-count and logical-qubit count. In particular, the authors estimated the total cost of the algorithm C as

7

$$C\simeq Q_{\mathcal{W}}\cdot C_{\mathcal{W}},$$ where $Q_{\mathcal{W}}$ and $C_{\mathcal{W}}$ are the query count of $\mathcal{W}$ and the cost per $\mathcal{W}$ , respectively. The costs of other parts of the algorithm are negligible and thus, not estimated in ref. ⁴. We show in Supplementary Note 1 how $Q_{\mathcal{W}}$ and $C_{\mathcal{W}}$ are determined.

In ref. ⁷, energy estimation is performed by applying QPE to the time-evolution operator e^−iHτ, which is approximated by the second-order Trotter formula, i.e.,

8

$$e^{-iH\tau }\approx {\left[\left(\prod _{l=1}^{K-1}{e}^{-i{H}_l\frac{\tau }{2r}}\right)e^{-i{H}_K\frac{\tau }{r}}\left(\prod _{l=K-1}^1{e}^{-i{H}_l\frac{\tau }{2r}}\right)\right]}^r,$$ where r is the number of Trotter steps, $H={\sum }_{l=1}^K{H}_l$ and H_l’s are Hermitian operators. Then, the total cost of the algorithm is

9

$$C\simeq Q_T\cdot C_T,$$ where Q_T and C_T are the query count of and cost per approximated e^−iHτ, respectively.

The FH Hamiltonian is divided into three H_l’s in ref. ⁷; H₁ = H_c, and H_h is decomposed into H₂ and H₃, each consisting of hopping terms that act on non-overlapping plaquettes, which are related by a simple lattice translation, as shown in Fig. 1a. This Trotter scheme comes with two main benefits. First, the translational symmetry between H₂ and H₃ simplifies, in practice, the evaluation of the Trotter error bounds²⁹. Second, every plaquette term can be diagonalized using Clifford gates and four two-site fermionic Fourier transforms⁵, turning $e^{-i{H}_{2/3}\theta }$ into a layer of same-angle R_z rotation gates. Instead of synthesizing the R_z gates individually, they are more efficiently synthesized collectively using a circuit optimization technique known as Hamming-weight phasing (HWP)^31,32. Furthermore, a chemical potential shift n ↦ n − 1/2 is applied to H_c, which leads to 3× fewer Pauli operators, arising from JW transformation⁵³, in H_c; the resulting change in the final energy estimate can be classically corrected⁷. Upon a Clifford transformation, $e^{-i{H}_c\theta }$ , a product of e^−iZZθ terms acting on disjoint pairs of qubits is reduced to a layer of same-angle R_z gates, which are again effected using HWP.

Acknowledgements

A.K. would like to acknowledge William Simon for his contribution on the F.H. simulations during the early stage of this work, Athena Caesura for optimizing the F.H. qubitization circuits from an earlier version of this manuscript, and Harriet Apel for spotting a few non-fatal errors, which did not affect the results, in the appendices of an earlier version of this manuscript.

Author contributions

B.S. wrote the code for evaluating the Trotter error bounds. A.K. designed the quantum circuits and performed the resource estimation. Both authors developed the Trotter schemes and wrote the paper.

Data availability

All data generated or analyzed during this study are included in this published article and its supplementary information files.

Code availability

The underlying code for this study is not publicly available but may be made available to qualified researchers on reasonable request from the corresponding author.

Competing interests

The authors declare no competing interests.

References

1.HubbardJ<article-title>Electron correlations in narrow energy bands III. An improved solutionProc. R. Soc. Lond. Ser. A. Math. Phys. Sci.19642814014191964RSPSA.281..401H2.LeBlancJPF<article-title>Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithmsPhys. Rev. X201550410413.WeckerD<article-title>Solving strongly correlated electron models on a quantum computerPhys. Rev. A2015920623182015PhRvA..92f2318W4.BabbushR<article-title>Encoding electronic spectra in quantum circuits with linear t complexityPhys. Rev. X201880410155.KivlichanID<article-title>Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via TrotterizationQuantum202042966.LemieuxJDuclos-CianciGSénéchalDPoulinD<article-title>Resource estimate for quantum many-body ground-state preparation on a quantum computerPhys. Rev. A20211030524082021PhRvA.103e2408L42725157.CampbellET<article-title>Early fault-tolerant simulations of the Hubbard modelQuantum Sci. Technol.202170150072022QS&T....7a5007C8.YoshiokaNOkuboTSuzukiYKoizumiYMizukamiW<article-title>Hunting for quantum-classical crossover in condensed matter problemsnpj Quantum Inf.2024102024npjQI..10...45Y459.LowGHChuangIL<article-title>Hamiltonian simulation by qubitizationQuantum2019316310.SuzukiM<article-title>General theory of fractal path integrals with applications to many-body theories and statistical physicsJ. Math. Phys.1991324004071991JMP....32..400S108836011.KivlichanID<article-title>Quantum simulation of electronic structure with linear depth and connectivityPhys. Rev. Lett.20181201105012018PhRvL.120k0501K377812712.Gidney, C. & Fowler, A. G. Efficient magic state factories with a catalyzed $|T⟩$ to 2 $\mid CCZ\left(\right.⟩$ transformation. Quantum3, 135 (2019).13.JiangH-CDevereauxTP<article-title>Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping t’Science2019365142414282019Sci...365.1424J396636714.QinM<article-title>Absence of superconductivity in the pure two-dimensional Hubbard modelPhys. Rev. X20201003101615.KimC<article-title>Systematics of the photoemission spectral function of cuprates: Insulators and hole- and electron-doped superconductorsPhys. Rev. Lett.199880424542481998PhRvL..80.4245K16.AndersenOLiechtensteinAJepsenOPaulsenF<article-title>LDA energy bands, low-energy Hamiltonians, t’, t”, t⊥(k), and J⊥J. Phys. Chem. Solids199556157315911995JPCS...56.1573A17.Delannoy, J.-Y. P., Gingras, M. J. P., Holdsworth, P. C. W. & Tremblay, A.-M. S. Low-energy theory of the t − t′ − t′′ − u Hubbard model at half-filling: Interaction strengths in cuprate superconductors and an effective spin-only description of La2CuO4. Phys. Rev. B. 79, 235130 (2009)..18.Dalla PiazzaB<article-title>Unified one-band Hubbard model for magnetic and electronic spectra of the parent compounds of cuprate superconductorsPhys. Rev. B2012851005082012PhRvB..85j0508D19.HirayamaMYamajiYMisawaTImadaM<article-title>Ab initio effective hamiltonians for cuprate superconductorsPhys. Rev. B2018981345012018PhRvB..98m4501H20.MoréeJ-BHirayamaMSchmidMTYamajiYImadaM<article-title>Ab initio low-energy effective hamiltonians for the high-temperature superconducting cuprates Bi₂Sr₂CuO₆, Bi₂Sr₂CaCu₂O₈, HgBa₂CuO₄, and CaCuO₂Phys. Rev. B20221062351502022PhRvB.106w5150M21.LebertBW<article-title>Paramagnon dispersion and damping in doped Na_xCa_2−xCuo₂Cl₂Phys. Rev. B20231080245062023PhRvB.108b4506L22.RaghuSQiX-LLiuC-XScalapinoDJZhangS-C<article-title>Minimal two-band model of the superconducting iron oxypnictidesPhys. Rev. B2008772205032008PhRvB..77v0503R23.MoreoADaghoferMRieraJADagottoE<article-title>Properties of a two-orbital model for oxypnictide superconductors: Magnetic order, B_2g spin-singlet pairing channel, and its nodal structurePhys. Rev. B2009791345022009PhRvB..79m4502M24.von BurgV<article-title>Quantum computing enhanced computational catalysisPhys. Rev. Res.2021303305525.LeeJ<article-title>Even more efficient quantum computations of chemistry through tensor hypercontractionPRX Quantum202120303052021PRXQ....2c0305L26.KimIH<article-title>Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte moleculesPhys. Rev. Res.2022402301927.Caesura, A. et al. Faster quantum chemistry simulations on a quantum computer with improved tensor factorization and active volume compilation https://arxiv.org/abs/2501.06165 (2025).28.Low, G. H. et al. Fast quantum simulation of electronic structure by spectrum amplification https://arxiv.org/abs/2502.15882 (2025).29.SchubertAMendlCB<article-title>Trotter error with commutator scaling for the fermi-hubbard modelPhys. Rev. B20231081951052023PhRvB.108s5105S30.WangQLiMMonroeCNamY<article-title>Resource-optimized fermionic local-hamiltonian simulation on a quantum computer for quantum chemistryQuantum2021550931.GidneyC<article-title>Halving the cost of quantum additionQuantum201827432.NamYMaslovD<article-title>Low-cost quantum circuits for classically intractable instances of the Hamiltonian dynamics simulation problemnpj Quantum Inf.201952019npjQI...5...44N4433.MullerDEPreparataFP<article-title>Bounds to complexities of networks for sorting and for switchingJ. ACM.19752219520138382934.WangGKanA<article-title>Option pricing under stochastic volatility on a quantum computerQuantum20248150435.KliuchnikovVLauterKMinkoRPaetznickAPetitC<article-title>Shorter quantum circuits via single-qubit gate approximationQuantum20237120836.Liu, W.-Y., Zhai, H., Peng, R., Gu, Z.-C. & Chan, G. K.-L. Accurate simulation of the Hubbard model with finite fermionic projected entangled pair states https://arxiv.org/abs/2502.13454 (2025).37.Gidney, C., Shutty, N. & Jones, C. Magic state cultivation: growing T states as cheap as CNOT gates https://arxiv.org/abs/2409.17595 (2024).38.Wan, K. H. Constant-time magic state distillation https://arxiv.org/abs/2410.17992 (2024).39.PoulinDKitaevASteigerDSHastingsMBTroyerM<article-title>Quantum algorithm for spectral measurement with a lower gate countPhys. Rev. Lett.20181210105012018PhRvL.121a0501P40.FomichevS<article-title>Initial state preparation for quantum chemistry on quantum computersPRX Quantum2024504033941.Berry, D. W. et al. Rapid initial state preparation for the quantum simulation of strongly correlated molecules https://arxiv.org/abs/2409.11748 (2024).42.DingZChenC-FLinL<article-title>Single-ancilla ground state preparation via LindbladiansPhys. Rev. Res.2024603314743.Štěpán Smíd, Meister, R., Berta, M. & Bondesan, R. Polynomial time quantum Gibbs sampling for fermi-hubbard model at any temperature https://arxiv.org/abs/2501.01412 (2025).44.Zhan, Y. et al. Rapid quantum ground state preparation via dissipative dynamics https://arxiv.org/abs/2503.15827 (2025).45.LeeS<article-title>Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistryNat. Commun.2023142023NatCo..14.1952L195246.SteudtnerM<article-title>Fault-tolerant quantum computation of molecular observablesQuantum20237116447.KurokiK<article-title>Unconventional pairing originating from the disconnected Fermi surfaces of superconducting LaFeAsO1-xfxPhys. Rev. Lett.20081010870042008PhRvL.101h7004K48.WatzenböckC<article-title>Photoexcitations in the Hubbard model: generalized Loschmidt amplitude analysis of impact ionization in small clustersPhys. Rev. B20221060851352022PhRvB.106h5135W49.Low, G. H. & Wiebe, N. Hamiltonian simulation in the interaction picture. https://arxiv.org/abs/1805.00675 (2019).50.Litinski, D. & Nickerson, N. Active volume: an architecture for efficient fault-tolerant quantum computers with limited non-local connections https://arxiv.org/abs/2211.15465 (2022).51.MetznerWVollhardtD<article-title>Correlated lattice fermions in d = ∞ dimensionsPhys. Rev. Lett.1989623243271989PhRvL..62..324M52.LiebEHWuFY<article-title>Absence of Mott transition in an exact solution of the short-range, one-band model in one dimensionPhys. Rev. Lett.196820144514481968PhRvL..20.1445L53.JordanPWignerE<article-title>Über das paulische ÄquivalenzverbotZ. f.ür. Phys.1928476316511928ZPhy...47..631J

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41534-025-01091-0.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Hubbard, J. Electron correlations in narrow energy bands III. An improved solution. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci.; 1964; 281, pp. 401-419.1964RSPSA.281.401H

2. LeBlanc, JPF et al. Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms. Phys. Rev. X; 2015; 5, 041041.

3. Wecker, D et al. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A; 2015; 92, 062318.2015PhRvA.92f2318W

4. Babbush, R et al. Encoding electronic spectra in quantum circuits with linear t complexity. Phys. Rev. X; 2018; 8, 041015.

5. Kivlichan, ID et al. Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via Trotterization. Quantum; 2020; 4, 296.

6. Lemieux, J; Duclos-Cianci, G; Sénéchal, D; Poulin, D. Resource estimate for quantum many-body ground-state preparation on a quantum computer. Phys. Rev. A; 2021; 103, 052408.2021PhRvA.103e2408L4272515

7. Campbell, ET. Early fault-tolerant simulations of the Hubbard model. Quantum Sci. Technol.; 2021; 7, 015007.2022QS&T..7a5007C

8. Yoshioka, N; Okubo, T; Suzuki, Y; Koizumi, Y; Mizukami, W. Hunting for quantum-classical crossover in condensed matter problems. npj Quantum Inf.; 2024; 10, 2024npjQI.10..45Y 45.

9. Low, GH; Chuang, IL. Hamiltonian simulation by qubitization. Quantum; 2019; 3, 163.

10. Suzuki, M. General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys.; 1991; 32, pp. 400-407.1991JMP..32.400S1088360

11. Kivlichan, ID et al. Quantum simulation of electronic structure with linear depth and connectivity. Phys. Rev. Lett.; 2018; 120, 110501.2018PhRvL.120k0501K3778127

12. Gidney, C. & Fowler, A. G. Efficient magic state factories with a catalyzed $|T⟩$ to 2 $\mid CCZ\left(\right.⟩$ transformation. Quantum3, 135 (2019).

13. Jiang, H-C; Devereaux, TP. Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping t’. Science; 2019; 365, pp. 1424-1428.2019Sci..365.1424J3966367

14. Qin, M et al. Absence of superconductivity in the pure two-dimensional Hubbard model. Phys. Rev. X; 2020; 10, 031016.

15. Kim, C et al. Systematics of the photoemission spectral function of cuprates: Insulators and hole- and electron-doped superconductors. Phys. Rev. Lett.; 1998; 80, pp. 4245-4248.1998PhRvL.80.4245K

16. Andersen, O; Liechtenstein, A; Jepsen, O; Paulsen, F. LDA energy bands, low-energy Hamiltonians, t’, t”, t⊥(k), and J⊥. J. Phys. Chem. Solids; 1995; 56, pp. 1573-1591.1995JPCS..56.1573A

17. Delannoy, J.-Y. P., Gingras, M. J. P., Holdsworth, P. C. W. & Tremblay, A.-M. S. Low-energy theory of the t − t′ − t′′ − u Hubbard model at half-filling: Interaction strengths in cuprate superconductors and an effective spin-only description of La2CuO4. Phys. Rev. B. 79, 235130 (2009).

18. Dalla Piazza, B et al. Unified one-band Hubbard model for magnetic and electronic spectra of the parent compounds of cuprate superconductors. Phys. Rev. B; 2012; 85, 100508.2012PhRvB.85j0508D

19. Hirayama, M; Yamaji, Y; Misawa, T; Imada, M. Ab initio effective hamiltonians for cuprate superconductors. Phys. Rev. B; 2018; 98, 134501.2018PhRvB.98m4501H

20. Morée, J-B; Hirayama, M; Schmid, MT; Yamaji, Y; Imada, M. Ab initio low-energy effective hamiltonians for the high-temperature superconducting cuprates Bi₂Sr₂CuO₆, Bi₂Sr₂CaCu₂O₈, HgBa₂CuO₄, and CaCuO₂. Phys. Rev. B; 2022; 106, 235150.2022PhRvB.106w5150M

21. Lebert, BW et al. Paramagnon dispersion and damping in doped Na_xCa_2−xCuo₂Cl₂. Phys. Rev. B; 2023; 108, 024506.2023PhRvB.108b4506L

22. Raghu, S; Qi, X-L; Liu, C-X; Scalapino, DJ; Zhang, S-C. Minimal two-band model of the superconducting iron oxypnictides. Phys. Rev. B; 2008; 77, 220503.2008PhRvB.77v0503R

23. Moreo, A; Daghofer, M; Riera, JA; Dagotto, E. Properties of a two-orbital model for oxypnictide superconductors: Magnetic order, B_2g spin-singlet pairing channel, and its nodal structure. Phys. Rev. B; 2009; 79, 134502.2009PhRvB.79m4502M

24. von Burg, V et al. Quantum computing enhanced computational catalysis. Phys. Rev. Res.; 2021; 3, 033055.

25. Lee, J et al. Even more efficient quantum computations of chemistry through tensor hypercontraction. PRX Quantum; 2021; 2, 030305.2021PRXQ..2c0305L

26. Kim, IH et al. Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules. Phys. Rev. Res.; 2022; 4, 023019.

27. Caesura, A. et al. Faster quantum chemistry simulations on a quantum computer with improved tensor factorization and active volume compilation https://arxiv.org/abs/2501.06165 (2025).

28. Low, G. H. et al. Fast quantum simulation of electronic structure by spectrum amplification https://arxiv.org/abs/2502.15882 (2025).

29. Schubert, A; Mendl, CB. Trotter error with commutator scaling for the fermi-hubbard model. Phys. Rev. B; 2023; 108, 195105.2023PhRvB.108s5105S

30. Wang, Q; Li, M; Monroe, C; Nam, Y. Resource-optimized fermionic local-hamiltonian simulation on a quantum computer for quantum chemistry. Quantum; 2021; 5, 509.

31. Gidney, C. Halving the cost of quantum addition. Quantum; 2018; 2, 74.

32. Nam, Y; Maslov, D. Low-cost quantum circuits for classically intractable instances of the Hamiltonian dynamics simulation problem. npj Quantum Inf.; 2019; 5, 2019npjQI..5..44N 44.

33. Muller, DE; Preparata, FP. Bounds to complexities of networks for sorting and for switching. J. ACM.; 1975; 22, pp. 195-201.383829

34. Wang, G; Kan, A. Option pricing under stochastic volatility on a quantum computer. Quantum; 2024; 8, 1504.

35. Kliuchnikov, V; Lauter, K; Minko, R; Paetznick, A; Petit, C. Shorter quantum circuits via single-qubit gate approximation. Quantum; 2023; 7, 1208.

36. Liu, W.-Y., Zhai, H., Peng, R., Gu, Z.-C. & Chan, G. K.-L. Accurate simulation of the Hubbard model with finite fermionic projected entangled pair states https://arxiv.org/abs/2502.13454 (2025).

37. Gidney, C., Shutty, N. & Jones, C. Magic state cultivation: growing T states as cheap as CNOT gates https://arxiv.org/abs/2409.17595 (2024).

38. Wan, K. H. Constant-time magic state distillation https://arxiv.org/abs/2410.17992 (2024).

39. Poulin, D; Kitaev, A; Steiger, DS; Hastings, MB; Troyer, M. Quantum algorithm for spectral measurement with a lower gate count. Phys. Rev. Lett.; 2018; 121, 010501.2018PhRvL.121a0501P

40. Fomichev, S et al. Initial state preparation for quantum chemistry on quantum computers. PRX Quantum; 2024; 5, 040339.

41. Berry, D. W. et al. Rapid initial state preparation for the quantum simulation of strongly correlated molecules https://arxiv.org/abs/2409.11748 (2024).

42. Ding, Z; Chen, C-F; Lin, L. Single-ancilla ground state preparation via Lindbladians. Phys. Rev. Res.; 2024; 6, 033147.

43. Štěpán Smíd, Meister, R., Berta, M. & Bondesan, R. Polynomial time quantum Gibbs sampling for fermi-hubbard model at any temperature https://arxiv.org/abs/2501.01412 (2025).

44. Zhan, Y. et al. Rapid quantum ground state preparation via dissipative dynamics https://arxiv.org/abs/2503.15827 (2025).

45. Lee, S et al. Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry. Nat. Commun.; 2023; 14, 2023NatCo.14.1952L 1952.

46. Steudtner, M et al. Fault-tolerant quantum computation of molecular observables. Quantum; 2023; 7, 1164.

47. Kuroki, K et al. Unconventional pairing originating from the disconnected Fermi surfaces of superconducting LaFeAsO1-xfx. Phys. Rev. Lett.; 2008; 101, 087004.2008PhRvL.101h7004K

48. Watzenböck, C et al. Photoexcitations in the Hubbard model: generalized Loschmidt amplitude analysis of impact ionization in small clusters. Phys. Rev. B; 2022; 106, 085135.2022PhRvB.106h5135W

49. Low, G. H. & Wiebe, N. Hamiltonian simulation in the interaction picture. https://arxiv.org/abs/1805.00675 (2019).

50. Litinski, D. & Nickerson, N. Active volume: an architecture for efficient fault-tolerant quantum computers with limited non-local connections https://arxiv.org/abs/2211.15465 (2022).

51. Metzner, W; Vollhardt, D. Correlated lattice fermions in d = ∞ dimensions. Phys. Rev. Lett.; 1989; 62, pp. 324-327.1989PhRvL.62.324M

52. Lieb, EH; Wu, FY. Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension. Phys. Rev. Lett.; 1968; 20, pp. 1445-1448.1968PhRvL.20.1445L

53. Jordan, P; Wigner, E. Über das paulische Äquivalenzverbot. Z. f.ür. Phys.; 1928; 47, pp. 631-651.1928ZPhy..47.631J