The experimental realization of increasingly complex quantum states underscores the pressing need for new methods of state learning and verification. In one such framework, quantum state tomography, the aim is to learn the full quantum state from data obtained by measurements. Without prior assumptions on the state, this task is prohibitively hard. Here, we present an efficient algorithm for learning states on n fermion modes prepared by any number of Gaussian and at most t non-Gaussian gates. By Jordan-Wigner mapping, this also includes n-qubit states prepared by nearest-neighbor matchgate circuits with at most t swap gates. Our algorithm is based exclusively on single-copy measurements and produces a classical representation of a state, guaranteed to be close in trace distance to the target state. The sample and time complexity of our algorithm is poly(n,2t); thus if t=O(log⁡(n)), it is efficient. We also show that, if t scales slightly more than logarithmically, any learning algorithm to solve the same task must be inefficient, under common cryptographic assumptions. We also provide an efficient property-testing algorithm that, given access to copies of a state, determines whether such a state is far or close to the set of states for which our learning algorithm works. In addition to the outputs of quantum circuits, our tomography algorithm is efficient for some physical target states, such as those arising in time dynamics and low-energy physics of impurity models. Beyond tomography, our work sheds light on the structure of states prepared with few non-Gaussian gates and offers an improved upper bound on their circuit complexity, enabling an efficient circuit-compilation method.

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.

We consider the problem of approximating the ground state energy of a fermionic Hamiltonian using a Gaussian state. In sharp contrast to the dense case [1, 2], we prove that strictly q-local sparse fermionic Hamiltonians have a constant Gaussian approximation ratio; the result holds for any connectivity and interaction strengths. Sparsity means that each fermion participates in a bounded number of interactions, and strictly q-local means that each term involves exactly q fermionic (Majorana) operators. We extend our proof to give a constant Gaussian approximation ratio for sparse fermionic Hamiltonians with both quartic and quadratic terms. With additional work, we also prove a constant Gaussian approximation ratio for the so-called sparse SYK model with strictly 4-local interactions (sparse SYK-4 model). In each setting we show that the Gaussian state can be efficiently determined. Finally, we prove that the O(n⁻¹/²) Gaussian approximation ratio for the normal (dense) SYK-4 model extends to SYK-q for even q > 4, with an approximation ratio of O(n¹/^2−q/4). Our results identify non-sparseness as the prime reason that the SYK-4 model can fail to have a constant approximation ratio [1, 2].

The challenge of preparing a system in a designated state spans diverse facets of quantum mechanics. To complete this task of steering quantum states, one can employ quantum control through a sequence of generalized measurements, which direct the system towards the target state. In an active version of this protocol, the obtained measurement readouts are used to adjust the protocol on the go. This enables a sped-up performance relative to the passive version of the protocol, where no active adjustments are included. In this work, we consider such active measurement-driven steering as applied to the challenging case of many-body quantum systems. The target states of highest interest would be those with multipartite entanglement. Such state preparation in a measurement-based protocol is limited by the natural constraints for system-detector couplings. We develop a framework for finding such physically feasible couplings, based on parent Hamiltonian construction. For helpful decision-making strategies, we offer Hilbert-space-orientation techniques, comparable to those used in navigation. The first one is to tie the active-decision protocol to the greedy accumulation of the cost function, such as the target state fidelity. We show the potential of a significant speedup, employing this greedy approach to a broad family of matrix product state targets. For system sizes considered here, an average value of the speedup factor f across this family settles about 20, for some targets even reaching a few thousands. We also identify a subclass of matrix product state targets, including the ground state of the Affleck-Kennedy-Lieb-Tasaki spin chain, for which the value of f increases with system size. In addition to the greedy approach, the second wayfinding technique is to map out the available measurement actions onto a quantum state machine. A decision-making protocol can be based on such a representation, using semiclassical heuristics. This state-machine-based approach can be applied to a more restricted set of targets, where it sometimes offers advantages over the cost-function-based method. We give an example of a W-state preparation, which is accelerated with this method by f≃3.5, outperforming the greedy protocol for this target.