Core Concepts

Quantum computation exhibits remarkable autonomy from classical complexity classes, as demonstrated by the existence of oracles that decouple the power of BQP from NP, PH, and other classical complexity classes.

Abstract

The paper explores the surprising behaviors of quantum polynomial-time (BQP) complexity relative to classical complexity classes like NP, PH, and PP. The authors leverage the breakthrough result of Raz and Tal, which showed an oracle relative to which BQP is not contained in PH, to prove a series of new oracle separations:
There exists an oracle relative to which NPBQP is not contained in BQPPH, resolving a longstanding open problem. This demonstrates a fundamental difference between classical randomness and quantum randomness, as classical randomness can be "pulled out" of algorithms in ways that quantum randomness cannot.
Conversely, there exists an oracle relative to which BQPNP is not contained in PHBQP.
Relative to a random oracle, PP (which equals PostBQP) is not contained in the "QMA hierarchy" - a quantum analogue of the polynomial hierarchy.
Relative to a random oracle, for every k, the k+1st level of the classical polynomial hierarchy is not contained in BQPΣP
k.
There exists an oracle where NP is contained in BQP, BQP equals P#P, and yet the polynomial hierarchy is infinite. This shows that if NP is contained in BQP, it need not imply a collapse of the polynomial hierarchy, in contrast to the classical case.
There exists an oracle where P=NP but BQP is strictly more powerful than P, demonstrating the autonomy of quantum computation from classical complexity.
The authors develop new techniques to achieve these results, including a "quantum-aware" random restriction lemma, a concentration theorem for the block sensitivity of AC0 circuits, and a provable analogue of the Aaronson-Ambainis conjecture for sparse oracles. These tools may be of independent interest.

Stats

There exists an oracle relative to which NPBQP is not contained in BQPPH.
There exists an oracle relative to which BQPNP is not contained in PHBQP.
Relative to a random oracle, PP is not contained in the "QMA hierarchy".
Relative to a random oracle, for every k, the k+1st level of the classical polynomial hierarchy is not contained in BQPΣP
k.
There exists an oracle where NP is contained in BQP, BQP equals P#P, and yet the polynomial hierarchy is infinite.
There exists an oracle where P=NP but BQP is strictly more powerful than P.

Quotes

"One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm."
"Quantum computation exhibits remarkable autonomy from classical complexity classes, as demonstrated by the existence of oracles that decouple the power of BQP from NP, PH, and other classical complexity classes."

Key Insights Distilled From

by Scott Aarons... at **arxiv.org** 04-26-2024

Deeper Inquiries

The techniques developed in this work can be applied to further understand the relationship between quantum and classical complexity classes in the unrelativized setting by providing insights into the limitations and capabilities of quantum computation compared to classical computation.
One way to extend these techniques is to explore the possibility of proving nonrelativizing results that establish the autonomy of quantum computation in scenarios where classical techniques cannot be applied. By identifying specific properties or characteristics of quantum algorithms that cannot be replicated by classical algorithms, researchers can gain a deeper understanding of the inherent differences between quantum and classical complexity classes.
Additionally, these techniques can be used to investigate the boundaries of quantum supremacy, particularly in scenarios where quantum algorithms exhibit significant advantages over classical algorithms. By studying the specific conditions under which quantum algorithms outperform classical algorithms, researchers can uncover new insights into the power and limitations of quantum computation in relation to classical complexity classes.

Beyond the explored properties in this work, there are several other complexity-theoretic properties that might exhibit a similar "autonomy" of quantum computation relative to classical computation. Some of these properties include:
Communication Complexity: Investigating the communication complexity of quantum protocols compared to classical protocols can reveal instances where quantum communication provides advantages that classical communication cannot replicate.
Circuit Complexity: Exploring the circuit complexity of quantum circuits versus classical circuits can uncover scenarios where quantum circuits exhibit unique properties that classical circuits do not possess.
Interactive Proof Systems: Studying the power of quantum interactive proof systems in contrast to classical interactive proof systems can highlight situations where quantum protocols offer distinct advantages in proof verification and complexity analysis.
Randomized Algorithms: Analyzing the performance of quantum randomized algorithms compared to classical randomized algorithms can reveal cases where quantum randomness leads to computational advantages that classical randomness cannot achieve.
By investigating these and other complexity-theoretic properties, researchers can further elucidate the autonomy of quantum computation and its distinct characteristics relative to classical computation.

There are computational problems and models where the separation between quantum and classical complexity classes can be shown to hold unconditionally, without the need for relativization. Some examples include:
Quantum Search Algorithms: Problems like Grover's algorithm for unstructured search showcase a provable quantum speedup over classical algorithms, demonstrating an unconditional separation between quantum and classical complexity classes in the context of search problems.
Quantum Fourier Transform: The efficiency and power of the Quantum Fourier Transform in quantum algorithms like Shor's algorithm for integer factorization provide instances where quantum computation surpasses classical computation without relying on relativization techniques.
Quantum Simulation: Problems related to quantum simulation, where quantum systems are simulated using quantum computers, can exhibit inherent separations between quantum and classical complexity classes due to the unique capabilities of quantum computers in simulating quantum phenomena.
By focusing on these types of computational problems and models, researchers can establish unconditional separations between quantum and classical complexity classes, shedding light on the intrinsic advantages of quantum computation in specific domains.

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