Efficient learning of quantum states prepared with few fermionic non-Gaussian gates (2402.18665v3)

Abstract: 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-neighbour 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 $\mathrm{poly}(n,2^t)$; 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.

• The paper introduces a polynomial-time algorithm that efficiently learns fermionic quantum states using single-copy measurements and trace distance guarantees.
• It compresses non-Gaussian effects into a few qubits via Gaussian operations, reducing the effective dimensionality and computational complexity.
• The study establishes optimal learnability boundaries and practical implications for quantum state tomography in impurity-driven physical systems.

Efficient Learning of Quantum States Prepared with Few Fermionic Non-Gaussian Gates

This paper presents novel methodologies in the field of quantum state tomography, specifically targeting the learning of states prepared by quantum circuits composed primarily of Gaussian gates with a limited number of non-Gaussian gates. The focus is directed towards systems involving fermionic modes, where the task at hand is the efficient characterization and learning of quantum state preparations under constraints of non-Gaussianity.

Theoretical Framework and Algorithm

The authors introduce an algorithm that efficiently learns states constructed within a framework that incorporates any number of Gaussian and at most $t$ non-Gaussian gates across $n$ fermion modes. This is accomplished by utilizing single-copy measurements and generating a classical representation that faithfully approximates the target state in terms of trace distance. The algorithm's complexity is polynomial in terms of $n$ and $2^t$; hence, it is notably efficient when $t = \mathcal{O}(\log(n))$.

The approach underlying the algorithm is grounded on a significant structural insight: any state within the given operational limits can be transformed via a Gaussian operation into a state where non-Gaussian effects are confined to a limited subsystem, specifically onto $\kappa t$ qubits where $\kappa$ is a small constant. This compression of non-Gaussianity, achieved through Gaussian operations, allows for reducing the effective dimensionality of the problem significantly, thereby simplifying the learning task.

Implications and Computational Complexity

The implications are twofold. Practically, it offers a method to perform efficient state tomography on quantum devices and circuits constrained by their reliance on Gaussian operations with sparse non-Gaussian intervention. Theoretically, it opens new insights into the structure and complexity of quantum states defined under these constraints.

Moreover, the paper rigorously determines boundaries for efficient learnability, demonstrated by establishing that if $t$ scales slightly more than logarithmically, based on standard cryptographic assumptions, then efficient learning becomes computationally infeasible. This confirms the exponential scaling in $t$ is necessary, thus affirming the optimal nature of their proposed algorithm’s dependence on $t$.

Physical Relevance and Extensions

The work extends beyond purely theoretical constructs and explores applications within physical systems, such as ground states arising from quantum impurity models. These states can be approximated by forms amenable to the proposed learning algorithm, thus exposing viable paths for experimental implementations in systems where impurity dynamics dominate, such as in cold atom setups.

Moreover, the work provides pathways for efficient circuit compilations, enabling quantum information processing tasks where circuit depth is critically constrained. The convergence of ideas from classical pseudorandomness into the quantum domain, mapped through qubit-to-fermion correspondences, further entices potential applications in quantum device security.

Concluding Remarks

This paper advances the field by clarifying the conditions under which certain quantum states can be efficiently learned. Through a careful blend of quantum computational theory and practical algorithm design, it sketches a clear landscape of possibilities and limitations, simultaneously offering efficient solutions and highlighting inherent challenges.

The outcomes hint promising applications in experimental quantum physics and quantum information science, potentially guiding future endeavors in the efficiency-driven deployment of quantum technologies.