Yihui Quek

In a variety of physically relevant settings for learning from quantum data, there is an established recipe for measuring polynomially many copies of that data such that the resulting measurement readouts contain enough information to reconstruct the underlying system. Yet designing protocols that can computationally efficiently extract that information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting and show a number of new information-computation gaps for quantum learning. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the ``standard'' variant of Learning Stabilizers with Noise and for agnostically learning product states.

We introduce Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm that utilizes coherent Bell measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian optimization to classical decoding. For a signed Pauli Hamiltonian $H$ and any degree-$\ell$ polynomial $\calP$, HDQI prepares a purification of the density matrix $$\rho_\calP(H) = \calP^2(H)/\Tr[\calP^2(H)]$$ by solving a combination of two tasks: decoding $\ell$ errors on a classical code defined by $H$, and preparing a pilot state that encodes the anti-commutation structure of $H$. Choosing $\calP(x)$ to approximate $\exp(-\beta x/2)$ yields Gibbs states at inverse temperature $\beta$; other choices of $\calP$ prepare approximate ground states, microcanonical ensembles, and other spectral filters. The decoding problem inherits structural properties of $H$; in particular, local Hamiltonians map to LDPC codes. Preparing the pilot state is always efficient for commuting Hamiltonians, but highly non-trivial for non-commuting Hamiltonians. Nevertheless, we prove that this state admits an efficient matrix product state representation for a class of nearly commuting Pauli Hamiltonians whose anti-commutation graph decomposes into connected components of logarithmic size. We show that HDQI efficiently prepares Gibbs states at arbitrary temperatures for a class of physically motivated commuting Hamiltonians -- including the toric code, color code, and Haah's cubic code -- but also develop a matching efficient classical algorithm for this task, thereby delineating the boundary of efficient classical simulation. For a non-commuting semiclassical spin glass and commuting stabilizer code Hamiltonians with quantum defects, HDQI provably prepares Gibbs states up to a constant inverse-temperature threshold using polynomial quantum resources and quasi-polynomial classical preprocessing. These results position HDQI as a versatile new algorithmic primitive, connecting quantum state preparation to classical decoding.

Random classical linear codes are widely believed to be hard to decode, exponentially so at constant coding rate. If the rate vanishes asymptotically sufficiently rapidly, slightly sub-exponential decoding algorithms are known. By contrast, the complexity of decoding a random quantum stabilizer code has remained an open question for quite some time. This work closes the gap in our understanding of the algorithmic hardness of decoding random quantum versus random classical codes. We prove that decoding a random stabilizer code with even a single logical qubit is at least as hard as decoding a random classical code at constant rate—the maximally hard regime. This result suggests that the easiest random quantum decoding problem is at least as hard as the hardest random classical decoding problem, and shows that any sub-exponential algorithm decoding a typical stabilizer code, at any coding rate, would immediately imply a breakthrough in cryptography. More generally, we also characterize many other complexity-theoretic properties of stabilizer codes. While classical decoding admits a random self-reduction, we prove significant barriers for the existence of random self-reductions in the quantum case. This result follows from new bounds on Clifford entropies and Pauli mixing times, which may be of independent interest. As a complementary result, we demonstrate various other self-reductions which are in fact achievable, such as between search and decision. Our work also demonstrates several ways in which quantum phenomena, such as quantum degeneracy, force several reasonable definitions of stabilizer decoding—all of which are classically identical—to have distinct or non-trivially equivalent complexity.

Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on quantum circuits. We first show that any noise `truncates' most quantum circuits to effectively logarithmic depth, in the task of computing Pauli expectation values. We then prove that quantum circuits under any non-unital noise exhibit lack of barren plateaus for cost functions composed of local observables. But, by leveraging the effective shallowness, we also design a classical algorithm to estimate Pauli expectation values within inverse-polynomial additive error with high probability over the ensemble. Its runtime is independent of circuit depth and it operates in polynomial time in the number of qubits for one-dimensional architectures and quasi-polynomial time for higher-dimensional ones. Taken together, our results showcase that, unless we carefully engineer the circuits to take advantage of the noise, it is unlikely that noisy quantum circuits are preferable over shallow quantum circuits for algorithms that output Pauli expectation value estimates, like many variational quantum machine learning proposals. Moreover, we anticipate that our work could provide valuable insights into the fundamental open question about the complexity of sampling from (possibly non-unital) noisy random circuits.