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ABSTRACT In classical and single-particle settings, non-trivial band topology always gives rise to robust boundary modes. For quantum many-body systems, however, multiple topological

fermions are not always able to coexist, since Pauli exclusion prevents additional fermions from occupying the limited number of available topological modes. In this work, we show, through

IBM quantum computers, how one can robustly stabilize more fermions than the number of topological modes through specially designed 2-fermion interactions. Our demonstration hinges on the

realization of BDI- and D-class topological Hamiltonians on transmon-based quantum hardware, and relied on a tensor network-aided circuit recompilation approach. We also achieved the full

reconstruction of multiple-fermion topological band structures through iterative quantum phase estimation (IQPE). All in all, our work showcases how advances in quantum algorithm

implementation enable noisy intermediate-scale quantum (NISQ) devices to be exploited for topological stabilization beyond the context of single-particle topological invariants. SIMILAR

CONTENT BEING VIEWED BY OTHERS REALIZATION OF HIGHER-ORDER TOPOLOGICAL LATTICES ON A QUANTUM COMPUTER Article Open access 10 July 2024 DIGITAL-ANALOG QUANTUM COMPUTING OF FERMION-BOSON

MODELS IN SUPERCONDUCTING CIRCUITS Article Open access 11 March 2025 QUANTUM ZENO MONTE CARLO FOR COMPUTING OBSERVABLES Article Open access 12 March 2025 INTRODUCTION Many-body quantum

effects like particle statistics and Hubbard interactions underscore some of the most exciting condensed matter phenomena, such as superconductivity and fractionalization1,2,3. Their

interplay with single-particle properties like band topology is particularly fascinating, with rich fractionalized quasiparticles emerging out of Coulomb repulsion within dispersionless

Landau levels, for instance. But unlike in purely single-body settings, particle statistics often play a central role, determining all Fermi surface properties like electrical conductivity.

In topological systems, Pauli exclusion also implies that topological robustness is only conferred upon the few electrons occupying the limited number of topological modes4. Compared to

single-body topological phenomena realizable in classical metamaterials, novel many-body phases are traditionally much harder to engineer and physically demonstrate. This is due to

challenges in accessing and manipulating intrinsically fragile quantum states, unlike the macroscopic classical degrees of freedom of photonic, acoustic and electrical circuit

lattices5,6,7,8. Fortunately, there is considerable recent experimental progress in various quantum systems such as ultracold atomic lattices,9, photonic systems10, silicon11 and trapped ion

systems12, which make Richard Feynman’s vision13 of utilizing quantum systems to simulate quantum Hamiltonians a reality. Of particular versatility are universal quantum simulators, also

known as quantum computers, which allow arbitrary quantum systems to be simulated, thereby in principle enabling any quantum phenomenon to be physically realized. Through the appropriate

design of sequences of quantum operations, known collectively as quantum algorithms, quantum computers are capable of performing a vast array of computational tasks, some at polynomial or

exponential resource advantage over classical algorithms. Indeed, even in the current noisy intermediate-scale quantum (NISQ) era, quantum computers have already shown great promise, with

demonstrations of quantum advantage in limited settings14,15, mapping topology in parameter space16, the achievement of chemical accuracy in intermediate-scale electronic structure

calculations17, and in neutron scattering and exotic magnetic phenomena simulations18,19. This work shall employ computations made with IBM quantum computers, which not only rank favourably

with other cutting-edge platforms20,21 in hardware performance, such as gate fidelities and decoherence times, but is also fully accessible via the cloud. IBM Quantum (IBM Q) currently

offers access to up to 65-qubit machines based on superconducting transmon qubits, and provides an open-source software development kit called _Qiskit_22,23. Thus far, IBM Q has been

successfully utilized in simulating spin models24, global quantum quenches25, quantum chemistry problems26, topological phenomena27,28,29,30, machine learning31 and various other

applications32. We emphasize that quantum computation in the current NISQ-era is still plagued with significant limitations. Main bottlenecks include low gate fidelity, decoherence, limited

qubit connectivity and limited number of qubits33, which together impose constraints on circuit depth and structure. Amidst qubit noise and readout error, typical depths of

\({{{\mathcal{O}}}}(1{0}^{1})\) entangling gate layers are presently feasible for precision results. Of utmost current priority is hence the development of error mitigation and circuit

optimization approaches that maximize computational capability within hardware bounds, thereby enabling the practical use of quantum simulators in contexts where classical simulators, for

example, purpose-designed electrical circuits34, are inadmissible. In this work, we stabilize BDI- and D-class topological boundary states, as well as demonstrate a full band structure

reconstruction of the fermionic extended Kitaev chain (KC) on IBM quantum computers. Compared to existing quantum computer realizations of other topological states27,28,29, some of which are

performed in parameter space, ours was performed on a longer (12-qubit) chain with physical open boundaries that host topological modes. Furthermore, our extended KC Hamiltonian contains

multiple non-local couplings which presented heavy demands on circuit depth and complexity, necessitating the use of tensor network-aided circuit recompilation techniques beyond traditional

trotterization. Crucially, by exploiting the quantum nature of the IBM machines, we engineered effective interactions to stabilize more fermions than originally allowed by the number of

topological zero modes, hence physically realizing a few-body phenomenon that has not been possible in existing classical realizations. RESULTS AND DISCUSSION PHYSICAL MOTIVATION AND MODELS

Topological robustness is a highly sought-after property exemplified by the extraordinarily long survival duration of boundary states in specially designed topological lattices35,36. This

robustness has inspired various potential applications like topological lasers and sensors37,38, and originates from the integer quantization of topological invariants characterizing the

lattice band structure4. While non-trivial topology has been demonstrated in a wide variety of classical metamaterials, true quantization of the response can only be observed in quantum

settings (but see refs. 39,40). Yet, a fully quantum topological fermionic system is also limited by the fact that there can only be as many topological fermions as available topological

modes. Since the latter is determined by the topological invariant, which is typically an integer of order \({{{\mathcal{O}}}}(1{0}^{0})\), it will be of great scientific and practical

interest to probe how quantum interactions can also _enhance_ the number of robustly surviving fermions beyond what is dictated by the topological invariant. We shall first introduce our

topological lattice models, and later discuss how specific interactions preserve the fidelity of multiple fermions by hosting many-body states that are adiabatically connected to

non-interacting topological states. In this work we simulate on a quantum computer the extended Kitaev model41,42, which is a 1D open chain containing next-nearest neighbour (NNN) couplings

in addition to its underlying nearest-neighbour (NN) structure, with two sites per unit cell: $$\begin{array}{ll}{H}^{{{{\rm{KC}}}}}=&\frac{1}{2}\left\{\mathop{\sum

}\limits_{j=1}^{N-1}\left[{v}_{1}\left({c}_{j}^{{\dagger} }{c}_{j+1}-{d}_{j}^{{\dagger} }{d}_{j+1}\right)+{{{\Delta }}}_{1}\left({d}_{j}^{{\dagger} }{c}_{j+1}-{c}_{j}^{{\dagger}

}{d}_{j+1}\right)\right]\right.\\ &+\mathop{\sum }\limits_{j=1}^{N-2}\left[{v}_{2}\left({c}_{j}^{{\dagger} }{c}_{j+2}-{d}_{j}^{{\dagger} }{d}_{j+2}\right)+{{{\Delta

}}}_{2}\left({e}^{i\phi }{d}_{j}^{{\dagger} }{c}_{j+2}-{e}^{-i\phi }{c}_{j}^{{\dagger} }{d}_{j+2}\right)\right]\\ &\left.+\mathop{\sum }\limits_{j=1}^{N}\mu \left({c}_{j}^{{\dagger}

}{c}_{j}-{d}_{j}^{{\dagger} }{d}_{j}\right)\right\}+{{{\rm{h.c.}}}},\end{array}$$ (1) for _N_ unit cells, chemical potential _μ_, tunnelling coefficients _v_1,2, superconducting pairing

Δ1,2, relative phase _ϕ_, and fermionic operators _c__j_, _d__j_ respectively acting on the A- and B-sublattices of the _j_-th unit cell. Our model _H_KC is an extension of the standard

Kitaev chain model that has been intensely studied for hosting Majorana zero modes43, which is of particular interest to fault-tolerant topological quantum computing44. Its extra degrees of

freedom in Eq. (1) allow for smooth tuning across the BDI- and D-symmetry protected topological (SPT) classes elaborated below, and will ultimately also be useful in the design of

interactions that preserve the robustness of multiple fermions. When _ϕ_ = 0, _H_KC preserves time-reversal symmetry (TRS) in addition to parity and charge conjugation symmetry, and belongs

to the BDI class of the tenfold-way topological classification45, characterized by a winding number \(\nu \in {\mathbb{Z}}\). Our particular model possesses _ν_ = 0, 1, 2 regimes, the first

trivial and the latter two respectively exhibiting twofold and fourfold degeneracy of midgap topological modes localized at either boundary. Mathematically, 2_π__ν_ corresponds to the

winding of its Berry phase across one Brillouin zone period (see Supplementary Note 1). When _ϕ_ ≠ 0, TRS is broken and _H_KC falls into the D-class characterized by \({{\mathbb{Z}}}_{2}\)

invariant, with its topologically trivial/non-trivial phases respectively labelled by Berry phase winding _γ_ = 0, _π_ which encapsulates the relative configurations of the _k_ and −_k_

paths41,42. This D-class phase minimally requires NNN couplings, not present in the BDI class which contains the extremely well-known Su-Schrieffer-Heeger (SSH) model46,47, and is thus much

less frequently investigated let alone realized in quantum settings48. We note that the SSH model, which we shall also simulate, is a special case of the extended Kitaev model upon unitary

rotation: $${H}^{{{{\rm{SSH}}}}}=\mathop{\sum }\limits_{j=1}^{N}v{c}_{j}^{{\dagger} }{d}_{j}+\mathop{\sum }\limits_{j=1}^{N-1}w{d}_{j}^{{\dagger} }{c}_{j+1}+{{{\rm{h.c.}}}},$$ (2) with _v_, 

_w_ intra-cell and inter-cell hopping coefficients. We remark that _H_SSH and _H_KC are quadratic in the fermionic basis and are, in principle, efficiently simulable classically49; but with

the addition of quartic interactions, both models cannot be efficiently classically simulated. PERSISTENT BOUNDARY MODES FROM TOPOLOGICAL PROTECTION As the first experiments on the IBM

quantum computer, we demonstrate that initial states at the boundary survive much longer when the Hamiltonian is topological. We emphasize that this is true for a rather large parameter

space of initial states, not just exact topological eigenstates (see Supplementary Note 4). For a start, various perfectly localized 1-fermion initial states defined in Table 1 are evolved

via _H_SSH and _H_KC. We utilize a tensor network-aided recompilation technique (see Methods) to construct quantum circuits for time-evolution, in order to overcome circuit depth limitations

on NISQ hardware. Our approach is based on prior literature50,51,52, with added sector-specificity to enhance performance at larger fermion numbers and qubit counts. Furthermore, we employ

a suite of error mitigation techniques to improve data quality, in particular readout mitigation (RO)25,53, post-selection (PS)25,54, and averaging across machines and qubit chains (see

Methods). Through computational-basis measurements on simulation qubits, the occupancy \({{{\bf{O}}}}={[\left\langle {O}_{1}\right\rangle \left\langle {O}_{2}\right\rangle \ldots

\left\langle {O}_{n}\right\rangle ]}^{\top }\) for \({O}_{j}={c}_{j}^{{\dagger} }{c}_{j}=(1-{\sigma }_{j}^{z})/2\) along the chain is retrieved, and the extent of evolution away from

\(\left|{\psi }_{0}\right\rangle\), whose occupancy is O0, can be assessed via the occupancy fidelity $${{{{\mathcal{F}}}}}_{O}={\left|{{{{\bf{O}}}}}^{\top }{{{{\bf{O}}}}}_{0}\right|}^{2}\in

[0,1].$$ (3) Note \({{{{\mathcal{F}}}}}_{O}\) is distinct from state fidelity \({{{\mathcal{F}}}}={\left|\left\langle {\psi }_{0}| \psi \right\rangle \right|}^{2}\)—see Supplementary Note 3

for a more detailed discussion. We present 1-particle time-evolution results for _H_SSH in Fig. 1a, comparing raw data, data with RO and PS mitigation, and data with both. The effectiveness

of the error mitigation methods is clear, with the occupancy fidelity of all initial states approaching close to exact diagonalization (ED) results with RO and PS applied. The stability,

_i.e_. persistence, of the initial state is quantified via the occupancy fidelity \({{{{\mathcal{F}}}}}_{O}\); the slower the decay in \({{{{\mathcal{F}}}}}_{O}\), the more stable the state.

As expected, the decay of the initial states \(\left|{1}^{{{{\rm{L}}}}}\right\rangle\), \(\left|{1}^{{{{\rm{R}}}}}\right\rangle\) localized at the left and right boundary sites are slow in

the topological regime, compared to that of the middle-localized state \(\left|{1}^{{{{\rm{M}}}}}\right\rangle\), which overlaps negligibly with topological boundary modes; by contrast, in

the trivial regime topological protection does not exist and all states decay quickly. We remark that edge-localized states can be stable even with significant perturbations, which is a

hallmark of topological robustness. For instance, the stability of \(\left|{1}^{{{{\rm{L}}}}}\right\rangle\), in the sense of slow decay of \({{{{\mathcal{F}}}}}_{O}\), is preserved even

when mixed with non-SPT states localized on the neighbouring and next-neighbouring sites (see Supplementary Note 4). For _H_KC with NNN couplings, even relatively delocalized initial states

can exhibit slow decay. We determine ‘idealized’ maximally localized boundary modes that are adiabatically connected to topological eigenstates (Supplementary Note 1), that exhibits

negligible decay, such as to facilitate the design of more general persistent states. In the BDI class, they are \(\left|{1}_{1}^{{{{\rm{L}}}}}\right\rangle\) and

\(\left|{1}_{2}^{{{{\rm{L}}}}}\right\rangle\) as defined in Table 1, which are localized on the first and second unit cells respectively. In Fig. 1c,

\(\left|{1}_{1}^{{{{\rm{L}}}}}\right\rangle\) (and \(\left|{1}_{1}^{{{{\rm{R}}}}}\right\rangle\)) are robust for both _ν_ = 1 and 2 topological sectors, but

\(\left|{1}_{2}^{{{{\rm{L}}}}}\right\rangle\) (and \(\left|{1}_{2}^{{{{\rm{R}}}}}\right\rangle\)) are robust only for the _ν_ = 2 sector, which supports two topological boundary modes at

each end. For D-class _H_KC systems with nonzero _ϕ_ and NNN couplings, the four idealized boundary states are each localized on the four boundary sites with _ϕ_-dependent relative phases

between site orbitals, as defined in Table 1. Indeed, almost negligible decay was observed (Fig. 1c) for these states in the topological regimes, compared to the middle-localized

\(\left|{1}^{{{{\rm{M}}}}}\right\rangle\) which do not benefit from topological robustness; in the trivial regime, all states decay quickly as expected. The extent of delocalization of

\(\left|{1}^{{{{\rm{M}}}}}\right\rangle\) in all regimes, and of all initial states in the topologically trivial regime, are detailed in the occupancy density maps of Fig. 1c. While state

initialization in the time-evolution circuits for _H_SSH and BDI-class _H_KC are performed through explicit circuit components (see Supplementary Note 3), that for

\(\left|{1}_{1,2}^{{{{\rm{L}}}},{{{\rm{R}}}}}\right\rangle\) of D-class _H_KC are absorbed into our dynamically optimized recompilation ansatz, so as to minimize incurred costs in circuit

depth. BAND STRUCTURE RECONSTRUCTION Our boundary states owe their slow decay to the gapped nature of their dominant (topological) eigenstate component, as is readily understood through a

two-component approximate treatment. Consider \(\left|\psi (t)\right\rangle =\alpha \left|{\psi }_{1}\right\rangle +\beta \left|{\psi }_{2}\right\rangle {e}^{-i{{\Delta }}Et}\) with

\({\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}=1\); then the state fidelity is \({{{\mathcal{F}}}}(t)={\left|\left\langle \psi (t)| \psi (0)\right\rangle

\right|}^{2}=1-4{\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}{\sin }^{2}({{\Delta }}Et/2)\), which stays near unity as long as \({\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}\ll

1\), that is, when a dominant eigenstate component exists. This can be checked by comparing all our initial states with exact topological eigenstates (Fig. 1b). In realistic settings where

the initial boundary state generically overlaps with arbitrarily many eigenstates, the energy gap between the dominant topological eigenstate and all other states is crucial to stability. It

introduces a separation of frequency scales that avoids overwhelming destructive interference. Experimentally, this energy gap can be verified from a full reconstruction of the topological

band structure. Below, we first describe our approach for mapping the band structure, and then present its results for both single-fermion and two-fermion systems. To probe and reconstruct

the band structure on quantum hardware, we perform iterative quantum phase estimation (IQPE)55,56, which supports the estimation of eigenenergies of a simulated Hamiltonian, in principle to

arbitrary precision (see Methods). Compared to quantum phase estimation (QPE)57,58, IQPE circuits are shallower and require fewer qubits, and are thus suited for implementation on

current-generation NISQ hardware. IQPE results for _H_SSH and _H_KC in the 1-particle sector are shown in Fig. 1d. The BDI topological phase transition in _H_SSH at _w_ = _v_ is apparent,

with a pair of midgap states separating from the bulk and approaching degeneracy at _E_ = 0 as _w_/_v_ increases. The _H_KC model possesses richer behaviour—in the BDI case, transitions from

the _ν_ = 1 phase into the trivial (_ν_ = 0) and then into _ν_ = 2 phase occur as _v_/_μ_ increases, for illustrative _v_ ≡ _v_1 = _v_2 = Δ1 = Δ2. The _ν_ = 1 phase exhibits twofold

degeneracy of midgap states like in the SSH model, and the _ν_ = 2 phase exhibits fourfold degeneracy at _E_ = 0 due to having two zero modes at each boundary. In the D class, twofold midgap

topological degeneracy occurs in the _γ_ = _π_ but not _γ_ = 0 phase. For the latter, fourfold degeneracy is broken as the eigenenergies split into pairs of positive energy and negative

energy states, which can be regarded as remnants of topological midgap states. In all cases, the reconstructed band structure from hardware execution of IQPE closely match ED results.

TOPOLOGICAL STABILITY FOR MULTIPLE FERMIONS—NON-INTERACTING CASE Compared to existing classical realizations of BDI- and D-class topological states, our IBM Q realizations possess the

advantage of granting natural access to quantum many-body effects like fermionic statistics and interactions. We shall first discuss the former, specifically on the various ways whereby up

to four fermions can simultaneously enjoy topological robustness. We first consider _H_SSH and the two-fermion sector, in which two-fermion boundary states can be constructed either as

\(\left|{2}^{{{{\rm{LR}}}}}\right\rangle =c{[{1}^{{{{\rm{L}}}}}]}^{{\dagger} }c{[{1}^{{{{\rm{R}}}}}]}^{{\dagger} }\left|{{{\rm{vac}}}}\right\rangle\) with 1 fermion at each boundary, or

alternatively as \(\left|{2}_{{{{\rm{AA}}}}}^{{{{\rm{L}}}}}\right\rangle\) or \(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{L}}}}}\right\rangle\) with one fermion in the

\(\left|{1}^{{{{\rm{L}}}}}\right\rangle\) orbital and the other in the nearest A or B site. Of these, only \(\left|{2}^{{{{\rm{LR}}}}}\right\rangle\) has both particles overlapping

significantly with topological boundary modes, so only it would be conferred stability in the topological phase (Fig. 2b). The \(\left|{2}_{{{{\rm{AA}}}}}^{{{{\rm{L}}}}}\right\rangle\) and

\(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{L}}}}}\right\rangle\) states, like \(\left|{1}^{{{{\rm{M}}}}}\right\rangle\), are unstable whether topological modes exist or not, despite one of the

fermions being in the topologically stable orbital \(\left|{1}^{{{{\rm{L}}}}}\right\rangle\). The upshot is that since _H_SSH possesses only one topological state at each end of the chain,

it is impossible to construct a 2-particle topological state localized at only one boundary. Generalization to multiple fermions follows. To realize 2-fermion topological states localized at

a _single_ boundary, the fourfold midgap topological modes of _H_KC, with two at each boundary, can be utilized. In particular, the two fermions can occupy

\(\left|{1}_{1}^{{{{\rm{L}}}}}\right\rangle\) and \(\left|{1}_{2}^{{{{\rm{L}}}}}\right\rangle\) near the left boundary, resulting in \(\left|{2}_{12}^{{{{\rm{L}}}}}\right\rangle\), or

analogously \(\left|{2}_{12}^{{{{\rm{R}}}}}\right\rangle\) for the right boundary. Relaxing the requirement of localization on a single edge, robust states such as \(|2_{ij}^{\rm{LR}}\rangle

=c{[1_i^{\rm{L}}]}^{\dagger }c{[1_j^{\rm{R}}]}^{\dagger} |{\rm{vac}}\rangle\) are all possible, where _i_, _j_ ∈ {1, 2}. Indeed, hardware results indicate these states are conferred

stability in the topological regime; the contrast against the non topologically protected middle-localized \(\left|{2}^{{{{\rm{M}}}}}\right\rangle\) is drastic (Fig. 2c). The occupancy

density maps concur that these 2-fermion edge localized states persist almost perfectly in the topological phase, but diffuses in the trivial phase. As a generalization to larger numbers of

effective particles, we demonstrate that up to 4 fermions can enjoy stability in the D-class _H_KC system, with each occupying a different topological mode. Indeed, the 4-fermion state

\(\left|{4}^{{{{\rm{LLRR}}}}}\right\rangle\) is robust compared to the middle-localized \(\left|{4}^{{{{\rm{M}}}}}\right\rangle\) state in the topological phase (Fig. 2c). STABILITY FOR

MULTIPLE FERMIONS—INTERACTING CASE Quantum interactions can present new phase transitions59 and avenues of stability with no non-interacting analogues60,61. The understanding of strongly

correlated topological models is also crucial in understanding the phenomenology of real materials62,63. While weak interactions may only perturb the stability of SPT boundary modes, strong

interactions can drastically break the symmetry protection altogether, leading to new preferred states. Our time-evolution and IQPE methods, hinging on circuit recompilation, readily

supports the study of strongly correlated models, facilitating digital quantum computers as alternative experimental platforms from existing ultracold atomic lattices64 or photonic

crystals65. We consider two-body Hubbard interactions which have been commonly used to model strong density-density interactions64,66. Among the simplest 1D topological model with

interactions is the SSH-Hubbard model, which has been explored in various contexts67,68. We are interested in Hubbard interactions that directly compete with the boundary-localization of SSH

topological modes. Specifically, we consider interactions between all successive sites (_U_2), sites within the same unit cell (_U_4), and same sublattice sites across adjacent unit cells

(_U_5). Our interacting SSH model is then obtained by adding these interaction terms to Eq. (2),

\({H}_{{{{\rm{full}}}}}^{{{{\rm{SSH}}}}}={H}^{{{{\rm{SSH}}}}}+{H}_{{{{\rm{int}}}}}^{{{{\rm{SSH}}}}}\), with

$$\begin{array}{ll}{H}_{{{{\rm{int}}}}}^{{{{\rm{SSH}}}}}=-\,({U}_{2}+{U}_{4})\mathop{\sum }\limits_{j=1}^{N}{c}_{j}^{{\dagger} }{c}_{j}{d}_{j}^{{\dagger} }{d}_{j}\\

\qquad\qquad-\,{U}_{2}\mathop{\sum }\limits_{j=1}^{N-1}{c}_{j+1}^{{\dagger} }{c}_{j+1}{d}_{j}^{{\dagger} }{d}_{j}-{U}_{5}\mathop{\sum }\limits_{j=1}^{N-1}{c}_{j+1}^{{\dagger}

}{c}_{j+1}{c}_{j}^{{\dagger} }{c}_{j},\end{array}$$ (4) where \({c}_{j}^{{\dagger} }\) (\({d}_{j}^{{\dagger} }\)) creates a fermion in the A (B) sublattice of the _j_-th unit cell. We

emphasize that these interactions are homogeneous across all unit cells, and do not induce boundary effects on their own, without the interplay with topological boundary localization. The

schematic in Fig. 3a illustrates the interaction terms in Eq. (4). The _U_4 and _U_5 terms are intentionally chosen to induce asymmetry across the two sublattices—they impact states with

inhomogeneous polarizations differently, such as SSH topological modes which occupy one sublattice exclusively. We dynamically evolve on IBM Q various boundary-localized 2-fermion states

\(\left|{2}^{{{{\rm{LR}}}}}\right\rangle\), \(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{L}}}}}\right\rangle\), \(\left|{2}_{{{{\rm{AA}}}}}^{{{{\rm{L}}}}}\right\rangle\),

\(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{R}}}}}\right\rangle\), \(\left|{2}_{{{{\rm{BB}}}}}^{{{{\rm{R}}}}}\right\rangle\), schematically shown in Fig. 3b. As before, the superscripts indicate the

edge localization (left/right) of the 2 fermions, while the subscripts indicate the occupied sublattices; we remind the reader that the chain starts with an A-site on the left. Our chosen

2-particle interactions have enabled us to achieve stable multi-fermion boundary states due to either topological localization or interaction-induced polarization, and often the interplay of

both. As such, we can for instance produce stable states localized on a single boundary in \({H}_{{{{\rm{full}}}}}^{{{{\rm{SSH}}}}}\), reminiscent of topological states, but yet existing

even in the topologically trivial regime of the non-interacting part _H_SSH. The conferred stability generally increases with interaction strength at diminishing returns; beyond _U_/_w_ ~ 

10, little additional stability is gained from stronger interactions. Yet, the impact of the topological character of the non-interacting terms in our systems remains significant even in the

face of arbitrarily strong interactions. In Fig. 3c–d, we present the evolution of various boundary-localized 2-fermion initial states subject to strong interactions _U_2, _U_4 or _U_5 of

relative strength 10. While such strong interactions may seem clearly dominant compared to the non-interacting _H_SSH part, the topological character of _H_SSH still affects the evolution of

most of these states significantly. When _U_2 is switched on (Fig. 3c), \(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{L}}}}}\right\rangle\) (and equivalently

\(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{R}}}}}\right\rangle\), not shown) retain near-perfect fidelity, even when the chain is topologically trivial. For the

\(\left|{2}^{{{{\rm{LR}}}}}\right\rangle\) initial state localized at both boundaries, topology enables a longer survival time. To obtain stable 2-fermion states localized at a single

boundary, we may switch on _U_4 and _U_5; when _U_5 > 0, both types of left boundary modes (\(\left|{2}_{{{{\rm{AB}}}}}^{{{{\rm{L}}}}}\right\rangle\) and

\(\left|{2}_{{{{\rm{AA}}}}}^{{{{\rm{L}}}}}\right\rangle\)) are stabilized. On the other hand, to obtain stable right boundary states, we require _U_4 > 0 as well, such as to balance with

_U_5. This is reflected from the stability plot in parameter space (Fig. 3e)—the time-averaged occupancy fidelity \(\overline{{{{{\mathcal{F}}}}}_{O}}\) for the left-localized 2-particle

boundary states is minimal along the _U_4 = _U_5 diagonal, being greatly destabilized relative to the right-localized modes. The preferential stabilization at either boundary by _U_4 and

_U_5 is a consequence of their coupling of specific sublattice pairs on finite chains; when the interaction couples all sublattices like for _U_2, two-fermion boundary modes are stable on

both edges, and the preferential stability is not observed. Earlier, we showed that the stability of initial boundary states can be maintained even when they differ rather significantly from

topological ansatz states (Supplementary Note 4). We now extend the same remark to boundary states that are not topologically protected, but which are conferred robustness under the

combination of topology and interaction-induced effects. In Fig. 3f, we mix \(\left|{2}_{{{{\rm{AA}}}}}^{{{{\rm{L}}}}}\right\rangle\) with

\(\left|{2}_{{{{\rm{BB}}}}}^{{{{\rm{L}}}}}\right\rangle\) perturbation, neither of which are SPT, and examine their evolution under varying _U_2 interactions. When _U_2 = 0, as expected,

\(\overline{{{{{\mathcal{F}}}}}_{O}}\) is relatively low; but for _U_2 > 0 an increase in \(\overline{{{{{\mathcal{F}}}}}_{O}}\) is observed, reaching ≳ 90% for _U_2 ≳ 5/2. This

stabilization holds across a wide range of mixing amplitudes. Most interestingly, the stabilization occurs only in the topological phase of _H_SSH—indeed in the trivial regime, _U_2 has

virtually no effect on state stability. Hence, neither topology nor interactions alone suffices to enable the observed perturbation-robust stabilization of non-SPT states. Similarly, one can

study Hubbard interactions on the extended Kitaev model. We again focus on interaction terms that induce sublattice asymmetry, and thus left/right boundary asymmetry for topological modes.

The extended Kitaev chain (Eq. (1)), however, has longer-range couplings compared to the SSH model (Eq. (2)). The NNN hoppings and pairings were crucial in the non-interacting model to

demonstrate a richer topology, exhibiting more than one pair of topological modes in the _ν_ = 2 phase of the BDI class. We thus consider interactions that compete with this topology,

specifically between occupied A-sites of NN (_U_2) an NNN (_U_3) unit cells, with \({H}_{{{{\rm{full}}}}}^{{{{\rm{KC}}}}}={H}^{{{{\rm{KC}}}}}+{H}_{{{{\rm{int}}}}}^{{{{\rm{KC}}}}}\),

$${H}_{{{{\rm{int}}}}}^{{{{\rm{KC}}}}}=-{U}_{2}\mathop{\sum }\limits_{j=1}^{N-1}{c}_{j}^{{\dagger} }{c}_{j}{c}_{j+1}^{{\dagger} }{c}_{j+1}-{U}_{3}\mathop{\sum

}\limits_{j=1}^{N-2}{c}_{j}^{{\dagger} }{c}_{j}{c}_{j+2}^{{\dagger} }{c}_{j+2},$$ (5) These interactions are schematically illustrated in Fig. 3a. In the BDI class of the non-interacting

part _H_KC, we dynamically evolve the various 2-fermion states used earlier, \(\left|{2}_{{{{\rm{AA}}}}}^{{{{\rm{L}}}}}\right\rangle\) and

\(\left|{2}_{{{{\rm{BB}}}}}^{{{{\rm{L}}}}}\right\rangle\), which now have large overlaps with the topological modes of the non-interacting Kitaev chain _H_KC (see Fig. 3b for schematics).

Recall that the BDI-class of _H_KC can host up to two pairs of topological modes at the two boundaries. As shown in Fig. 4a, the presence of either interaction (_U_2 or _U_3) worsens the

fidelity of all 2-particle boundary modes when _ν_ = 2. These interactions hence disrupt the symmetry protection conferred in the BDI class. When _ν_ = 1, there is one fewer pair of

topological boundary modes, but the states \(\left|{2}_{{{{\rm{A0A}}}}}^{{{{\rm{L}}}}}\right\rangle\) and \(\left|{2}_{{{{\rm{B0B}}}}}^{{{{\rm{R}}}}}\right\rangle\)—which do not overlap

significantly with any non-interacting topological mode—now enjoy enhanced robustness in the presence of interactions. Also, all edge-localized modes can be made robust in the topologically

trivial case _ν_ = 0. In the D class of _H_KC, we demonstrate that _U_3 interactions can confer stability to 3- and 4- fermion states (Fig. 4b), which carry a greater number of fermions on a

boundary than available topological modes, and are hence topologically unprotected. The stability of SPT \(\left|{2}_{{{{\rm{12}}}}}^{{{{\rm{L}}}}}\right\rangle\) is not destroyed when _U_3

is imposed, and simultaneously the non-SPT \(\left|{3}^{{{{\rm{LLL}}}}}\right\rangle\) and \(\left|{4}^{{{{\rm{LLLR}}}}}\right\rangle\) are stabilized, in both the _γ_ = 0 topological and

trivial phases of _H_KC, the latter more pronounced. Though not shown in the main text, _U_2 has a similar but weaker effect (see Supplementary Note 4). We are thus able to engineer, in both

_H_SSH and (both BDI- and D-classes of) _H_KC, enhanced stability of edge modes that goes beyond what can be achieved from standard topology alone. These results demonstrate how NNN Hubbard

interactions can lead to avenues of interesting dynamical behaviour, which is not wholly surprising, given that they are known to lead to new phases like topological Mott phases in other

contexts imbued with appropriate lattice structure, for example, the honeycomb lattice69. Finally, we present the full two-fermion band structure of the interacting SSH model (Eq. (4))

reconstructed on quantum hardware using IQPE (Fig. 3g). Similar results are obtained against _U_5 instead of _U_2 (Supplementary Note 4). With strong interactions, an additional band of

energies whose large gap scales linearly with interaction strength appears. However, the interpretation of this additional set of bands is not straightforward. The interacting Hamiltonian

does not share the same eigenstates as its non-interacting counterpart; in fact differences in eigenstate wavefunctions are drastic even for modest _U_ ~ _v_, _w_ comparable to the

non-interacting _H_SSH. One therefore cannot hope to understand the separation of the band purely using the non-interacting eigenstates. Moreover, ED reveals that the band does not

necessarily contain 2-fermion boundary-localized modes, even though initial 2-fermion boundary-localized states can be stabilized in some cases. The near-perfect fidelity of these stabilized

2-particle boundary modes, some representatives having been previously demonstrated (Fig. 3), is due to strong overlap with one of the many eigenstates \(\left|\psi \right\rangle\) of the

interacting Hamiltonian; but the location of \(\left|\psi \right\rangle\) in the band structure may be in either set of bands. DISCUSSION AND OUTLOOK By realizing the interacting extended

Kitaev Chain on IBM quantum computers, we have demonstrated how various boundary states can be robustly preserved by topological mechanisms of BDI- and D-class symmetries. Importantly, this

topological protection is not limited to topological eigenstates, and interplays non-trivially with 2-body interactions and Pauli exclusion statistics in multi-fermion settings. Most

spectacularly, we discovered avenues where interactions allow more fermions to be stabilized at one boundary than suggested by the number of available topological modes alone. Our work also

illustrates how tensor network-aided circuit recompilation techniques beyond traditional trotterization enable the simulation and full band structure reconstruction of complex topological

Hamiltonians on quantum circuits. Our approach can be further extended to more sophisticated many-body interacting Hamiltonians, presenting new opportunities and raising the state-of-the-art

in the quantum simulation of strongly correlated topological systems. METHODS QUANTUM SIMULATION OF STATE EVOLUTION Given an initial state \(\left|{\psi }_{0}\right\rangle\) and a

time-independent Hamiltonian _H_ ∈ {_H_SSH, _H_KC}, the time-evolved state is \(\left|\psi (t)\right\rangle =U(t)\left|{\psi }_{0}\right\rangle\), with propagator _U_(_t_) = _e_−_i__H__t_. A

schematic of the quantum circuit performing time-evolution is given in Fig. 5a. Traditional implementation of _U_(_t_) on quantum circuits entails expanding the Hamiltonian _H_ in the

spin-1/2 basis and employing trotterization70,71; but acceptable trotterization error requires numerous time steps, yielding infeasibly deep circuits. Furthermore, each trotter step,

comprising terms \({e}^{-i{\beta }_{{{{\boldsymbol{\sigma }}}}}{{{\boldsymbol{\sigma }}}}{{\Delta }}t}\) for Pauli strings Σ and coefficients _β_Σ (see Supplementary Note 3), requires layers

of entangling gates scaling with the weight of Σ, thus impairing its usefulness for models with longer-range couplings. To transcend these limitations, we employ an implementation strategy

known as circuit recompilation, built upon prior studies50,51,52. A circuit ansatz (Fig. 5c) comprising _n__L_ ≤ 8 repetitive layers of general _U_3 rotation and CX entangling gates is

iteratively optimized, through tensor network-based quantum simulation72, to approach the intended unitary. Specifically, the _U_3 angles Χ = (Θ, Φ, Λ) are fine-tuned with L-BFGS-B with

basin-hopping (see Supplementary Note 3). We design the circuit ansatz to contain no longer-range CXs, in order to conform to qubit connectivity on hardware, and the CXs are placed in a

brickwork pattern to maximize entangling power, to accommodate the intended unitary in as few layers as possible. This technique yields order-of-magnitude shallower circuits than

trotterization at comparable error rates, and is critical in our acquisition of high-quality experiment data on NISQ hardware. For larger fermion numbers (≥3), we improve the optimization

procedure to focus on the relevant sectors, referred to as sector-specific recompilation, so as to overcome the impairment in performance brought about by the increased entanglement in

evolved states. We employ error mitigation techniques to improve the quality of hardware results—first, readout error mitigation (RO) reverses bit-flips during measurement based on prior

calibration of qubits25,53; second, post-selection (PS) is performed on particle number25,54. Indeed, since \({H}_{{{{\rm{full}}}}}^{{{{\rm{SSH}}}}}\) and

\({H}_{{{{\rm{full}}}}}^{{{{\rm{KC}}}}}\) are number-conserving, time-evolved states that fall outside the particle number sector of \(\left|{\psi }_{0}\right\rangle\) are unphysical and can

be discarded; no additional circuit depth is incurred since the measurements for O suffice. To feasibly accommodate the number of qubits _n_ used in our simulations (7 ≤ _n_ ≤ 13), we adopt

a tensored RO scheme that differs from the open-source implementation in _Qiskit_22. See Supplementary Note 2 for an introductory overview of quantum computation, and Supplementary Note 3

for technical details of our abovementioned quantum simulation methods. We end the section with a conceptual remark pertaining to the mapping between the fermionic models and the spin-1/2

qubits on the quantum computers. In traditional trotterization, the Jordan-Wigner or Bravyi-Kitaev transforms70,71 map the fermionic Hamiltonian into spin chains; the constituent Pauli

strings are then naturally expressible on quantum circuits. In comparison, using circuit recompilation, instead of assembling quantum circuits from a fixed set of formulae, an approximate

mapping between the fermionic propagator and spin-1/2 quantum gates is dynamically determined during optimization. ITERATIVE QUANTUM PHASE ESTIMATION Given \(U\left|\psi \right\rangle

={e}^{2\pi i\phi }\left|\psi \right\rangle\) for unitary _U_ and eigenstate \(\left|\psi \right\rangle\), IQPE estimates the eigenphase _ϕ_ ∈ [0, 1), in principle to arbitrary precision.

Setting _U_ = _e_−_i__H__t_ allows the inference of eigenenergy _E_ = −2_π__ϕ_/_t_ of \(\left|\psi \right\rangle\). As mentioned, compared to QPE57,58, IQPE circuits are shallower and

require fewer qubits—only a single ancilla qubit and a single controlled-unitary block is required. There is no need for multi-qubit inverse Fourier transforms. Truncating the binary

expansion _ϕ_ = 0. _ϕ_1_ϕ_2…_ϕ__m_ to _m_ bits, IQPE iterates from _k_ = _m_ to _k_ = 1; in iteration _k_, a controlled-\({U}^{{2}^{k-1}}\) block and a feedback _R__z_(_ω__k_) = 

−2_π_(0.0_ϕ__k_+1_ϕ__k_+2…_ϕ__m_) rotation are applied, and the ancilla qubit is measured to determine _ϕ__k_. An IQPE circuit diagram is shown in Fig. 5b. To minimize circuit depth, the

initialization of \(\left|\psi \right\rangle\) and the controlled-unitary block are implemented via recompilation; RO mitigation is applied to all qubits, and PS is applied to the simulation

qubits to select for specific particle number sectors. Indeed, by performing IQPE over numerous \(\left|\psi \right\rangle\)—which need not be exact eigenstates, since superposition states

collapse at measurement and yield expected eigenenergies nonetheless—the band structure of the Hamiltonian _H_ can be recovered at arbitrarily high resolution. See Supplementary Note 3 for

further implementation details. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. CODE AVAILABILITY The

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Google Scholar  Download references ACKNOWLEDGEMENTS J.M.K. thanks Shi-Ning Sun of Caltech for helpful discussions on quantum computing and algorithms. We acknowledge the use of IBM Quantum

services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team. FUNDING This work is supported by MOE

Tier 1 grant 21-0048-A0001-0. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, 91125, USA Jin

Ming Koh * Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge, CB3 0HE, UK Tommy Tai * Department of Physics, National University of Singapore, Singapore, 117542,

Singapore Yong Han Phee, Wei En Ng & Ching Hua Lee * School of Computing, National University of Singapore, Singapore, 117417, Singapore Wei En Ng Authors * Jin Ming Koh View author

publications You can also search for this author inPubMed Google Scholar * Tommy Tai View author publications You can also search for this author inPubMed Google Scholar * Yong Han Phee View

author publications You can also search for this author inPubMed Google Scholar * Wei En Ng View author publications You can also search for this author inPubMed Google Scholar * Ching Hua

Lee View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS C.H.L. initiated and supervised the project, and wrote parts of the manuscript. J.M.K.

developed most of the quantum simulation codebase, ran experiments on emulators and hardware, and wrote most of the manuscript. T.T. contributed to the codebase, ran experiments, and wrote

parts of the manuscript. Y.H.P. and W.E.N. provided support on emulators and hardware usage, and edited the manuscript. All authors analyzed computational and experiment results. The

manuscript reflects the contributions of all authors. CORRESPONDING AUTHORS Correspondence to Jin Ming Koh or Ching Hua Lee. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no

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permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Koh, J.M., Tai, T., Phee, Y.H. _et al._ Stabilizing multiple topological fermions on a quantum computer. _npj Quantum Inf_ 8, 16 (2022).

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