Soumik Ghosh

While quantum computing can accomplish tasks that are classically intractable, the presence of noise may destroy this advantage in the absence of fault tolerance. In this work, we present a quasi-polynomial-time classical algorithm for simulating quantum circuits under local depolarization noise, thereby ruling out their quantum advantage in these settings. Our algorithm leverages a property called approximate Markov property to sequentially sample from the measurement outcome distribution of noisy circuits. We establish approximate Markov property in a broad range of circuits: (1) we prove that it holds for any circuit when the noise rate exceeds a constant threshold, and (2) we provide strong analytical and numerical evidence that it holds for random quantum circuits subject to any constant noise rate, including non-unital noises. These regimes include previously known classically simulable cases as well as new ones, such as shallow random circuits and random circuits under non-unital noise, where anticoncentration does not hold and prior algorithms fail. Taken together, our results significantly extend the boundary of classical simulability and suggest that noise generically enforces approximate Markov property and classical simulability, thereby highlighting the limitation of noisy quantum circuits in demonstrating quantum advantage.

We present a sample- and time-efficient algorithm to learn any bosonic Fock state acted upon by an arbitrary Gaussian unitary. As a special case, this algorithm efficiently learns states produced in Fock state BosonSampling, thus resolving an open question put forth by Aaronson and Grewal (Aaronson, Grewal 2023). We further study a hierarchy of classes of states beyond Gaussian states that are specified by a finite number of their higher moments. Using the higher moments, we find a full spectrum of invariants under Gaussian unitaries, thereby providing necessary conditions for two states to be related by an arbitrary (including active, e.g.~beyond linear optics) Gaussian unitary.

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.

Deep thermalization refers to the emergence of Haar-like randomness from quantum systems upon partial measurements. As a generalization of quantum thermalization, it is often associated with high complexity and entanglement. Here, we introduce computational deep thermalization and construct the fastest possible dynamics exhibiting it at infinite effective temperature. Our circuit dynamics produce quantum states with low entanglement in polylogarithmic depth that are indistinguishable from Haar random states to any computationally bounded observer. Importantly, the observer is allowed to request many copies of the same residual state obtained from partial projective measurements on the state --- this condition is beyond the standard settings of quantum pseudorandomness, but natural for deep thermalization.

We show that concrete hardness assumptions about learning or cloning the output state of a random quantum circuit can be used as the foundation for secure quantum cryptography. In particular, under these assumptions we construct secure one-way state generators (OWSGs), digital signature schemes, quantum bit commitments, and private key encryption schemes. We also discuss evidence for these hardness assumptions by analyzing the best-known quantum learning algorithms, as well as proving black-box lower bounds for cloning and learning given state preparation oracles. Our random circuit-based constructions provide concrete instantiations of quantum crypto- graphic primitives whose security do not depend on the existence of one-way functions. The use of random circuits in our constructions also opens the door to NISQ-friendly quantum cryp- tography. We discuss noise tolerant versions of our OWSG and digital signature constructions which can potentially be implementable on noisy quantum computers connected by a quantum network. On the other hand, they are still secure against noiseless quantum adversaries, raising the intriguing possibility of a useful implementation of an end-to-end cryptographic protocol on near-term quantum computers. Finally, our explorations suggest that the rich interconnections between learning theory and cryptography in classical theoretical computer science also extend to the quantum setting.

We study a counterintuitive property of ‘conditioning’ on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system (A,B) drawn from a k-design, we measure subsystem B in the computational basis, keep the outcome and examine the state that remains in subsystem A, approximating the overall ensemble (the ‘projected ensemble') by a k’-design. We ask: how does the design quality change due to this procedure, or how does k’ compare to k? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a k=O(1) design, the projected ensemble is Haar-random (or a k' = infinity design) in the thermodynamic limit (i.e. when the size of subsystem B is infinite). This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Finally, we show that if the global state is a k-design, with no assumption on how it was generated, the projected ensemble on subsystem A is a k/2 design. This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.

Quantum mechanics provides cryptographic primitives whose security is grounded in hardness assumptions independent of those underlying classical cryptography. However, existing proposals require low-noise quantum communication and long-lived quantum memory, capabilities which remain challenging to realize in practice. In this work, we introduce a quantum digital signature scheme that operates with only classical communication, using the classical shadows of states produced by random circuits as public keys. We provide theoretical and numerical evidence supporting the conjectured hardness of learning the private key (the circuit) from the public key (the shadow). A key technical ingredient enabling our scheme is an improved state-certification primitive that achieves higher noise tolerance and lower sample complexity than prior methods. We realize this certification by designing a high-rate error-detecting code tailored to our random-circuit ensemble and experimentally generating shadows for 32-qubit states using circuits with ≥ 80 logical (≥ 582 physical) two-qubit gates, attaining 0.90±0.01 fidelity. With increased number of measurement samples, our hardware-demonstrated primitives realize a proof-of-principle quantum digital signature, demonstrating the near-term feasibility of our scheme.

Quantum computational pseudorandomness has emerged as a fundamental notion that spans connections to complexity theory, cryptography and fundamental physics. However, all known constructions of efficient quantum-secure pseudorandom objects rely on complexity theoretic assumptions. In this work, we establish the first unconditionally secure efficient pseudorandom constructions against shallow-depth quantum circuit classes. We prove that: 1. Any quantum state $2$-design yields unconditional pseudorandomness against both $\QNC^0$ circuits with arbitrarily many ancillae and $\AC^0\circ\QNC^0$ circuits with nearly linear ancillae. 2. Random phased subspace states, where the phases are picked using a $4$-wise independent function, are unconditionally pseudoentangled against the above circuit classes. 3. Any unitary $2$-design yields unconditionally secure parallel-query pseudorandom unitaries against geometrically local $\QNC^0$ adversaries, even with limited $\AC^0$ postprocessing. Our results stand in stark contrast to the standard guarantee of the $2$-design property, which only ensures that they cannot be distinguished from Haar random ensembles using two copies or queries. Our work demonstrates that quantum computational pseudorandomness can be achieved unconditionally for natural classes of restricted adversaries, opening new directions in quantum complexity theory.