Solving interacting fermionic quantum many-body problems as they are ubiquitous in quantum chemistry and materials science is a central task of theoretical and numerical physics, a task that can commonly only be addressed in the sense of providing approximations of ground states. For this reason, it is important to have tools at hand to assess how well simple ansatzes would fare. In this work, we provide rigorous guarantees on how well fermionic Gaussian product states can approximate the true ground state, given a weighted interaction graph capturing the interaction pattern of the systems. Our result can be on the one hand seen as a extendibility result of fermionic quantum states: It says in what ways fermionic correlations can be distributed. On the other hand, this is a non-symmetric de-Finetti theorem for fermions, as the direct fermionic analog of a theorem due to Brandao and Harrow. We compare the findings with the distinctly different situation of distinguishable finite-dimensional quantum systems, comment on the approximation of ground states with Gaussian states and elaborate on the connection to the no low-energy trivial state conjecture.
Solving interacting fermionic quantum many-body problems as they are ubiquitous in quantum chemistry and materials science is a central task of theoretical and numerical physics, a task that can commonly only be addressed in the sense of providing approximations of ground states. For this reason, it is important to have tools at hand to assess how well simple ansatzes would fare. In this work, we provide rigorous guarantees on how well fermionic Gaussian product states can approximate the true ground state, given a weighted interaction graph capturing the interaction pattern of the systems. Our result can be on the one hand seen as a extendibility result of fermionic quantum states: It says in what ways fermionic correlations can be distributed. On the other hand, this is a non-symmetric de-Finetti theorem for fermions, as the direct fermionic analog of a theorem due to Brandao and Harrow. We compare the findings with the distinctly different situation of distinguishable finite-dimensional quantum systems, comment on the approximation of ground states with Gaussian states and elaborate on the connection to the no low-energy trivial state conjecture.
A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order D−1/2 for D-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a k-copy Gaussian state that is n-exchangeable under graded fermionic copy permutations is O(k/n)-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as O(r−1), and hence, by recoverability, the state admits an O(r−1/2) trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.
One of the main problems in computational physics is predicting the low-energy behavior of many-body quantum systems. The computational complexity of this problem, however, is relatively poorly understood. A recent major progress in this direction has been the no low-energy trivial states (NLTS) theorem; it gives a family of qubit Hamiltonians whose low-energy states cannot be reached by shallow quantum circuits. In this work we provide a fermionic counterpart to this theorem, constructing local fermionic Hamiltonians with no low-energy trivial states. Distinct from the qubit case, we define trivial states via finite-depth fermionic quantum circuits. We further strengthen the result, allowing free access to (generally, deep) Gaussian fermionic circuits into our notion of a trivial state. The desired fermionic Hamiltonian can be constructed using any qubit Hamiltonian which has the NLTS property via well-spread distributions over bitstrings. We also define a fermionic analog of quantum probabilistically checkable proofs (PCPs) and explore the relation of fermionic PCP class with the qubit version.
Quantum versions of de Finetti’s theorem are powerful tools, yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing one to bound the error made by mean-field approaches. Such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles. In this work, we derive a de Finetti theorem for finite sized Majorana fermionic systems. It is shown, much reflecting the spirit of other quantum de Finetti theorems, that a state which is invariant under certain permutations of modes loses most of its anti-symmetric character and is locally well described by a mode separable state. We discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of the Hartree-Fock approximation of quantum systems. We hint at a link to generalized Pauli principles for one-body reduced density operators. Finally, building upon the obtained de Finetti theorem, we generalize and extend the applicability of Hudson’s fermionic central limit theorem.
We investigate (k_{1},k_{2}) extendibility of fermionic Gaussian states, a property central to quantum correlations and approximations of separability. We show that these states are (k_{1},k_{2})-extendible if and only if they admit a fermionic Gaussian extension, yielding a complete covariance-matrix characterization and a simple semidefinite program (SDP) whose size scales linearly with the number of modes. This provides necessary conditions for arbitrary fermionic states and is sufficient within the Gaussian setting. Our main result is a finite de Finetti-type theorem: we derive trace-norm bounds between (k_{1},k_{2})-extendible fermionic Gaussian states and separable states, improving previous exponential scaling to linear in the number of modes, with complementary relative entropy and squashed entanglement bounds. For two modes, upper and lower bounds match at order 1/sqrt[k_{1}k_{2}]. Extendibility also provides operational support for one of the different notions of separability in fermionic systems. Finally, for fermionic Gaussian channels, we provide a simple semidefinite program criterion for antidegradability and show that entanglement-breaking channels coincide with replacement channels, implying no nontrivial entanglement-breaking fermionic Gaussian channels exist.
Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.
Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.