The Complexity of NISQ (2210.07234v1)

Abstract: The recent proliferation of NISQ devices has made it imperative to understand their computational power. In this work, we define and study the complexity class $\textsf{NISQ} $, which is intended to encapsulate problems that can be efficiently solved by a classical computer with access to a NISQ device. To model existing devices, we assume the device can (1) noisily initialize all qubits, (2) apply many noisy quantum gates, and (3) perform a noisy measurement on all qubits. We first give evidence that $\textsf{BPP}\subsetneq \textsf{NISQ}\subsetneq \textsf{BQP}$, by demonstrating super-polynomial oracle separations among the three classes, based on modifications of Simon's problem. We then consider the power of $\textsf{NISQ}$ for three well-studied problems. For unstructured search, we prove that $\textsf{NISQ}$ cannot achieve a Grover-like quadratic speedup over $\textsf{BPP}$. For the Bernstein-Vazirani problem, we show that $\textsf{NISQ}$ only needs a number of queries logarithmic in what is required for $\textsf{BPP}$. Finally, for a quantum state learning problem, we prove that $\textsf{NISQ}$ is exponentially weaker than classical computation with access to noiseless constant-depth quantum circuits.

• The paper formally defines the NISQ complexity class for hybrid classical-quantum systems, distinguishing it from classical and fault-tolerant quantum computing.
• Oracle separations provide theoretical evidence positioning NISQ strictly between the BPP and BQP complexity classes.
• Analysis of specific problems shows NISQ achieves advantages on some tasks like Bernstein-Vazirani but faces limitations on others like Unstructured Search and Shadow Tomography due to noise.

The Complexity of NISQ

This paper addresses the complexity class $NISQ$, designed to capture the computational capabilities of hybrid classical-quantum systems operating in the noisy intermediate-scale quantum (NISQ) era. The core focus is to understand where $NISQ$ sits in relation to established complexity classes such as $BPP$ and $BQP$. The authors provide evidence that $NISQ$ is a new class that expands beyond classical problems solvable in $BPP$ while remaining more constrained than $BQP$, which includes all problems solvable by quantum computers.

Key Contributions

1. Definition and Structure of $NISQ$: The researchers formally define $NISQ$ as the class of all problems that can be efficiently solved by a classical computer coupled with a noisy quantum device. The quantum device operates under limitations: preparing an initial noisy state, executing noisy quantum gates, and performing noisy measurements.
2. Oracle Separations:

The paper presents oracle separations as evidence for the positioning of $NISQ$ between $BPP$ and $BQP$: - $BPP \subsetneq NISQ$: A modified version of Simon's problem demonstrates a super-polynomial speedup using a robustified oracle that $NISQ$ can leverage effectively versus classical computation. - $NISQ \subsetneq BQP$: An alternate version shows that $NISQ$ is exponentially less powerful than fault-tolerant quantum computation relative to this oracle.

1. Analysis of Specific Problems:

The authors use three well-studied problems to further explore $NISQ$'s capabilities: - Unstructured Search: Outlines that $NISQ$ algorithms do not achieve Grover's quadratic speedup, needing $\Omega(N)$ queries. - Bernstein-Vazirani: Demonstrates that $NISQ$ uses only $\mathcal{O}(\log n)$ queries, a significant quantum advantage. - Shadow Tomography: Shows an exponential separation from $BQP$, requiring $(1-\lambda)^{-n}$ samples due to noise constraints.

Theoretical Implications

The paper puts forward that $NISQ$ represents a novel, intermediate complexity class distinct from both classical and fault-tolerant quantum computing. The highlighted separations provide a foundational understanding of how noisy quantum systems could expand the problem space beyond classical computation. However, there remain clear limitations due to the noise that undermine more expansive quantum advantages.

Practical Implications and Future Directions

Practically, the paper suggests that while NISQ-era devices may not match the power of error-corrected quantum computers, they still hold potential for certain problems. For future developments:

• Tightening the bounds on $NISQ$ against classical computation (e.g., exponential rather than super-polynomial separations) remains an open question.
• Exploring specific quantum algorithms (e.g., Shor's, Forrelation) within $NISQ$ devices could provide further insights into the utility of these systems.
• Understanding how spatial locality constraints affect $NISQ$ could illuminate more realistic operational scenarios.

This work lays a theoretical framework for analyzing quantum systems in the NISQ era and suggests methodologies for approaching future quantum devices that fall short of full fault tolerance.