Kishor Bharti

Pseudorandom quantum states (PRSs) and pseudorandom unitaries (PRUs) possess the dual nature of being efficiently constructible while appearing completely random to any efficient quantum algorithm. In this study, we establish fundamental bounds on pseudorandomness. We show that PRSs and PRUs exist only when the probability that an error occurs is negligible, ruling out their generation on noisy intermediate-scale and early fault-tolerant quantum computers. Further, we show that PRUs need imaginarity while PRS do not have this restriction. This implies that quantum randomness requires in general a complex-valued formalism of quantum mechanics, while for random quantum states real numbers suffice. Additionally, we derive lower bounds on the coherence of PRSs and PRUs, ruling out the existence of sparse PRUs and PRSs. We also show that the notions of PRS, PRUs and pseudorandom scramblers (PRSSs) are distinct in terms of resource requirements. We introduce the concept of pseudoresources, where states which contain a low amount of a given resource masquerade as high-resource states. We define pseudocoherence, pseudopurity and pseudoimaginarity, and identify three distinct types of pseudoresources in terms of their masquerading capabilities. Our work also establishes rigorous bounds on the efficiency of property testing, demonstrating the exponential complexity in distinguishing real quantum states from imaginary ones, in contrast to the efficient measurability of unitary imaginarity. Lastly, we show that the transformation from a complex to a real model of quantum computation is inefficient, in contrast to the reverse process, which is efficient. Our results establish fundamental limits on property testing and provide valuable insights into quantum pseudorandomness.

Decoders are a critical component of fault-tolerant quantum computing. They must identify errors based on syndrome measurements to correct quantum states. While finding the optimal correction is NP-hard and thus extremely difficult, approximate decoders with faster runtime often rely on uncontrolled heuristics. In this work, we propose a family of hierarchical quantum decoders with a tunable trade-off between speed and accuracy while retaining guarantees of optimality. We use the Lasserre Sum-of-Squares (SOS) hierarchy from optimization theory to relax the decoding problem. This approach creates a sequence of Semidefinite Programs (SDPs). Lower levels of the hierarchy are faster but approximate, while higher levels are slower but more accurate. We demonstrate that even low levels of this hierarchy significantly outperform standard Linear Programming relaxations. Our results on rotated surface codes and honeycomb color codes show that the SOS decoder approaches the performance of exact decoding. We find that Levels 2 and 3 of our hierarchy perform nearly as well as the exact solver. We analyze the convergence using rank-loop criteria and compare the method against other relaxation schemes. This work bridges the gap between fast heuristics and rigorous optimal decoding.

In adversarial settings, where attackers can deliberately and strategically corrupt quantum data, standard quantum error correction reaches its limits. It can only correct up to half the code distance and must output a unique answer. Quantum list decoding offers a promising alternative. By allowing the decoder to output a short list of possible errors, it becomes possible to tolerate far more errors, even under worst-case noise. But two fundamental questions remain: which quantum codes support list decoding, and can we design decoding schemes that are secure against efficient, computationally bounded adversaries? In this work, we answer both. To identify which codes are list-decodable, we provide a generalized version of the Knill-Laflamme conditions. Then, using tools from quantum cryptography, we build an unambiguous list decoding protocol based on pseudorandom unitaries. Our scheme is secure against any quantum polynomial-time adversary, even across multiple decoding attempts, in contrast to previous schemes. Our approach connects coding theory with complexity-based quantum cryptography, paving the way for secure quantum information processing in adversarial settings.

The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Here, we have shown that any bipartite pure entangled state can be self-tested through Quantum Steering. Analogous to the tilted CHSH inequality, we use a steering inequality called Tilted Steering Inequality for self-testing any pure two-qubit entangled state. We have further used this inequality to self-test any bipartite pure entangled state by certifying two-dimensional sub-spaces of the qudit state by observing the structure of the set of assemblages obtained on the trusted side after measurements are made on the un-trusted side. Finally, as a novel feature of self testing via steering, we use the notion of Assemblage based Robust Self Testing to provide robustness bounds for the self testing result in the case of pure maximally entangled states of any local dimension.