Shantanav Chakraborty

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 develop new algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework that encapsulates most known quantum algorithms and serves as the foundation for new ones. Existing implementations of QSVT rely on block encoding, incurring an intrinsic $O(\log L)$ ancilla overhead and circuit depth $\widetilde{O}(L d\lambda )$ for polynomial transformations of a Hamiltonian $H=\sum_{k=1}^L H_k$, where $d$ is the polynomial degree and $\lambda=\sum_{k}\|H_k\|$. We introduce a simple yet powerful approach that utilizes only basic Hamiltonian simulation techniques, namely, Trotter methods, to: (i) eliminate the need for block encoding, (ii) reduce the ancilla overhead to only a single qubit, and (iii) still maintain near-optimal complexity. Our method achieves a circuit depth of $\widetilde{O}(L(d\lambda_{\mathrm{comm}})^{1+o(1)})$, without requiring any complicated multi-qubit controlled gates. Moreover, $\lambda_{\mathrm{comm}}$ depends on the nested commutators of the terms of $H$ and can be substantially smaller than $\lambda$ for many physically relevant Hamiltonians, a feature absent in standard QSVT. To achieve these results, we make use of Richardson extrapolation in a novel way, systematically eliminating errors in any interleaved sequence of arbitrary unitaries and Hamiltonian evolution operators, thereby establishing a general framework that encompasses QSVT but is more broadly applicable. We further design two randomized algorithms for QSVT in settings with only sampling access to the Hamiltonian terms. The first is a direct randomization of standard QSVT, while the second integrates qDRIFT within our interleaved-circuit architecture. Both achieve a complexity quadratic in $d$, which we establish as a lower bound for any randomized method implementing polynomial transformations in this model. Finally, as applications, we develop end-to-end quantum algorithms for solving linear systems and estimating ground state properties of Hamiltonians, both achieving near-optimal complexity without relying on oracular access. Overall, our results establish a new framework for quantum algorithms, significantly reducing hardware overhead while maintaining near-optimal performance, with implications for both near-term and fault-tolerant quantum computing.