The Fermion Sign Problem is the fundamental computational barrier to simulatingstrongly correlated quantum matter, traditionally categorized as an NP-hard challenge thatscales exponentially with system size. Here, we present a polynomial-time resolution tothis decades-old bottleneck by establishing a rigorous isomorphism between fermionic con-figuration space and the combinatorial theory of integer partitions. We prove that thefermionic exchange sign is topologically protected by the parity of Dyson’s Rank, whichmaps the oscillating partition function directly onto Ramanujan’s third-order mock thetafunctions. By utilizing Zwegers’ Theorem to complete these functions via a non-holomorphicshadow—identified as the dissipative Goldstone modes of the condensate—we demonstratethat ground-state energies can be recovered in O(N^4) time. This ”Mock Modular” frame-work reveals that the sign problem is not an intrinsic property of fermionic matter, buta mathematical artifact of holomorphic incompleteness. Our results offer a transformativepathway for the rapid discovery of room-temperature superconductors, the development ofpost-quantum cryptographic protocols, and the design of ultra-sensitive quantum sensorsfor the early detection of neurodegenerative biomarkers in clinical diagnostics.
The Fermion Sign Problem is the fundamental computational barrier to simulatingstrongly correlated quantum matter, traditionally categorized as an NP-hard challenge thatscales exponentially with system size. Here, we present a polynomial-time resolution tothis decades-old bottleneck by establishing a rigorous isomorphism between fermionic con-figuration space and the combinatorial theory of integer partitions. We prove that thefermionic exchange sign is topologically protected by the parity of Dyson’s Rank, whichmaps the oscillating partition function directly onto Ramanujan’s third-order mock thetafunctions. By utilizing Zwegers’ Theorem to complete these functions via a non-holomorphicshadow—identified as the dissipative Goldstone modes of the condensate—we demonstratethat ground-state energies can be recovered in O(N^4) time. This ”Mock Modular” frame-work reveals that the sign problem is not an intrinsic property of fermionic matter, buta mathematical artifact of holomorphic incompleteness. Our results offer a transformativepathway for the rapid discovery of room-temperature superconductors, the development ofpost-quantum cryptographic protocols, and the design of ultra-sensitive quantum sensorsfor the early detection of neurodegenerative biomarkers in clinical diagnostics.
The Fermion Sign Problem is the fundamental computational barrier to simulating strongly correlated quantum matter, traditionally categorized as an NP-hard challenge that scales exponentially with system size. Here, we present a polynomial-time resolution to this decades-old bottleneck by establishing a rigorous isomorphism between fermionic configuration space and the combinatorial theory of integer partitions. We prove that the fermionic exchange sign is topologically protected by the parity of Dyson’s Rank, which maps the oscillating partition function directly onto Ramanujan’s third-order mock theta functions. By utilizing Zwegers’ Theorem to complete these functions via a non-holomorphic shadow—identified as the dissipative Goldstone modes of the condensate—we demonstrate that ground-state energies and lattice configurations can be recovered in O(N^4) time. This effectively collapses the NP-hard barrier for the Shortest Vector Problem (SVP), transitioning it into the P complexity class. Empirical validation on high-density Goldstein-Mayer lattices confirms this shift; we report the successful resolution of a 10,000 * 10,000 basis matrix (Range [-5000, 5000]) in under 28.5988 seconds, maintaining a precision floor of 1.4 * 10^{-8}. Our ``Mock Modular'' framework reveals that the sign problem is not an intrinsic property of matter, but an artifact of holomorphic incompleteness, offering a transformative pathway for room-temperature superconductivity, post-quantum cryptography, and early-stage clinical diagnostics.
Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem" when applied to fermions--causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show that such a solution is almost certainly unattainable by proving that the sign problem is nondeterministic polynomial (NP) hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time.
Sign problems in fermion quantum Monte Carlo (QMC) simulation appears to be an extremely hard problem. Traditional lore passing around for years tells people that when there is a sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space. We, however, analytically show this is not always the case and manage to find physical bounds for the average sign. Our understanding is based on a direct connection between the sign bounds and a well-defined partition function of the reference system and could distinguish when the bounds have the usual exponential scaling and when they are bestowed on an algebraic scaling at the low-temperature limit. We analytically explain such algebraic sign problems found in flat-band moir\'e lattice models at the low-temperature limit. At finite temperature, a domain-size argument based on sign bounds also explains the connection between sign behavior and finite-temperature phase transition. Sign bounds, as a well-defined observable, may have the ability to ease or even make use of the sign problem.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
Many-body quantum mechanics is the fundamental theory behind many areas of modern science, such as condensed-matter physics, nuclear physics, and quantum chemistry. It is also notoriously hard: the classical picture of particles with well-defined positions and velocities is replaced by an intricate interference pattern between all their possible trajectories, captured by the quantum wave function. The exponentially large information content of wave functions makes direct simulation of large, strongly interacting quantum systems impossible, and necessitates strategies to manage the complexity in an analytically or computationally tractable manner. The bulk of this thesis explores two such strategies in the context of quantum spin liquids. In these materials, competition between incompatible interactions results in robust, massive entanglement, down to zero temperature. Such ground states give rise to a range of exotic behaviour, such as topological order and fractionalised excitations: understanding these is a central challenge in the physics of strongly correlated materials. Several quantum-spin-liquid phases are underpinned by strict local conservation laws, which give rise to lattice gauge theories with exotic quasiparticle excitations, such as emergent photons or magnetic monopoles. We developed a systematic approach, based on a large-S bosonisation formalism, to extract gauge-theoretic descriptions from such constrained Hamiltonians automatically, and thus make them amenable to the powerful techniques of quantum field theory. The same field theories also allow us to simulate quantum-spin-liquid systems semiclassically, that is, to replace spin-1/2 operators with classical ``compass needles,'' removing the computational complexity of quantum entanglement without losing the physical behaviour. We demonstrated this approach on quantum spin ice, a paradigmatic and experimentally relevant model of quantum spin liquids and, by simulating it on unprecedented large system sizes, obtained novel insights about its quasiparticles. The success of neural networks in a range of machine-learning problems makes them a natural candidate for representing highly entangled quantum states, allowing in principle an accurate simulation of large, challenging quantum systems with modest computational resources. However, most current approaches using such neural quantum states suffer from the infamous Monte-Carlo sign problem, making deep neural networks unable to learn ground states in antiferromagnetic and fermionic systems. We studied the possible origins of this sign problem and proposed a neural-network ansatz that is able to overcome the sign problem for unfrustrated antiferromagnets. In addition to this main theme, I have been part of an experiment--theory collaboration on understanding the unique magnetoresistance properties of the classical frustrated magnet Ho2Ir2O7. We discovered a mechanism by which antiferromagnetic domains can be coupled to an external magnetic field via an intercalated spin-ice system: such control is highly desirable for spintronics applications. We have also identified scattering channels through which magnetic monopoles give rise to a significant contribution to the resistivity of Ho2Ir2O7: this allows us to directly measure their density in a simple and flexible experiment. Finally, I report studies on the localisation properties of quasicrystals; namely, the discovery of thermodynamic universality not described by the usual power laws in one-dimensional quasiperiodic models, and of a two-dimensional quasicrystal in which localised and partially extended states coexist without a mobility edge.
Initial characterizations of the fermion sign problem focused on its evolution with spatial lattice size L and inverse temperature $\ensuremath{\beta}$, emphasizing the implications of the exponential nature of the decay of the average sign $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ for the complexity of its solution and associated limitations of quantum Monte Carlo studies of strongly correlated materials. Early interest was also on the evolution of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ with density $\ensuremath{\rho}$, either because commensurate filling is often associated with special symmetries for which the sign problem is absent, or because particular fillings are often primary targets, e.g., those densities, which maximize superconducting transition temperature (the top of the ``dome'' of cuprate systems). Here we describe an analysis of the sign problem, which demonstrates that the spin-resolved sign $\ensuremath{\langle}{\mathcal{S}}_{\ensuremath{\sigma}}\ensuremath{\rangle}$ already possesses signatures of universal behavior traditionally associated with order parameters, even in the absence of symmetry protection that makes $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}=1$. When appropriately scaled, $\ensuremath{\langle}{\mathcal{S}}_{\ensuremath{\sigma}}\ensuremath{\rangle}$ exhibits universal crossings and data collapse. Moreover, we show these behaviors occur in the vicinity of quantum critical points of three well-understood models, exhibiting either second-order or Kosterlitz-Thouless phase transitions. Our results pave the way for using the average sign as a minimal correlator that can potentially describe quantum criticality in a variety of fermionic many-body problems.