Research & Teaching - Uttam Singh

2. Quantum Computation

Scalable quantum computation, capable of outperforming classical systems for a wide range of tasks, fundamentally relies on quantum error correction (QEC) to maintain coherence and mitigate errors caused by environmental noise and imperfect operations. However, the technical challenges associated with implementing fault-tolerant QEC at scale remain significant, making fully scalable quantum computation beyond the reach of current technologies. In this context, the intermediate regime of low-depth quantum computation, characteristic of the Noisy Intermediate-Scale Quantum (NISQ) era, has emerged as a promising and accessible paradigm for near-term exploration.

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Our research focuses on the following key aspects of low-depth quantum computation:

Understanding the Power of Low-Depth Quantum Computation: Low-depth quantum circuits, which involve fewer layers of quantum gates, are inherently less susceptible to noise and decoherence, making them suitable for current quantum devices. We aim to rigorously study the computational capabilities of such circuits, investigating problems they can efficiently solve and identifying tasks where they may surpass classical counterparts. This includes exploring quantum algorithms tailored to shallow-depth constraints and analyzing the role of entanglement and coherence in enhancing their power.

Verification of Quantum Computation: When quantum devices become increasingly complex and involve untrusted parties, ensuring their correctness becomes critically important. However, the problem of efficiently verifying quantum computations, particularly those in the complexity class BQP (Bounded-Error Quantum Polynomial Time), using classical computational resources (BPP, or Bounded-Error Probabilistic Polynomial Time), has remained an open challenge for a long time. Although fully scalable, fault-tolerant quantum computers are currently beyond the reach of quantum technology, an intriguing question arises: Can low-depth quantum computations, which involve relatively shallow quantum circuits, be verified using correspondingly low-depth classical computations? Addressing this problem would enable practical verification methods for near-term quantum devices operating within the constraints of noisy intermediate-scale quantum (NISQ) technologies. This would not only advance our understanding of quantum-classical interactions but also provide a pathway for certifying the results of quantum computations in real-world applications.

Computational Notions of Quantum Resources: The success of quantum computation relies on harnessing and managing quantum resources like coherence, entanglement, and magic states. We seek to formalize the computational role of these resources in low-depth settings, identifying how they contribute to the power of quantum circuits. This involves developing resource-theoretic frameworks that connect physical constraints with computational capabilities, ultimately providing a deeper understanding of the trade-offs between resource consumption and computational performance.

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