Complexity-Theoretic Foundations of Quantum Supremacy Experiments (1612.05903v2)

Abstract: In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by the the Quantum AI group at Google. We show that there's a natural hardness assumption, which has nothing to do with sampling, yet implies that no efficient classical algorithm can pass a statistical test that the quantum sampling procedure's outputs do pass. Compared to previous work, the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, for simulating a general quantum circuit with n qubits and m gates in polynomial space and m^O(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem--of the form "if approximate quantum sampling is classically easy, then PH collapses"--must be non-relativizing. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that the Fourier Sampling problem achieves a constant versus linear separation between quantum and randomized query complexities. Fifth, we study quantum supremacy relative to oracles in P/poly. Previous work implies that, if OWFs exist, then quantum supremacy is possible relative to such oracles. We show that some assumption is needed: if SampBPP=SampBQP and NP is in BPP, then quantum supremacy is impossible relative to such oracles.

• The paper introduces the Heavy Output Generation problem and QUATH assumption, establishing the classical intractability of sampling from quantum circuits.
• It develops a polynomial-space simulation algorithm that effectively models grid-based quantum circuits with optimized depth considerations.
• The authors demonstrate that non-relativizing techniques and black-box models are essential for proving the maximal separation in quantum versus classical query complexities.

Overview of "Complexity-Theoretic Foundations of Quantum Supremacy Experiments"

This essay provides an expert summary of the paper "Complexity-Theoretic Foundations of Quantum Supremacy Experiments" by Scott Aaronson and Lijie Chen, which explores the theoretical underpinnings for experiments that aim to demonstrate quantum supremacy using special-purpose quantum computers. These proposed experiments involve quantum systems, notably circuits of 40-50 high-quality qubits, to outperform classical computers significantly on specific tasks.

The paper explores major theoretical challenges and advances in demonstrating quantum supremacy via sampling problems rather than traditional decision or function problems. It underscores the necessity of achieving a quantum advantage, which potentially challenges the Extended Church-Turing Thesis (ECT), asserting that any physical process can be simulated by a Turing machine with polynomial overhead.

Key Contributions

The paper is built around five central results that address various challenges and implications of quantum supremacy experiments:

1. Hardness of Sampling from Quantum Circuits:
   • The paper introduces the Heavy Output Generation (HOG) problem, calling for a classical algorithm that could generate outputs with high quantum probabilities from a given quantum circuit. A key assumption, QUATH (Quantum Threshold Assumption), suggests the exponential difficulty for classical algorithms to solve HOG, thereby supporting the hardness of sampling problems.
2. New Algorithms for Simulating Quantum Circuits:
   • By developing a polynomial-space algorithm akin to Savitch’s Theorem, the authors propose efficient simulation methods for general and grid-based quantum circuits. This algorithm operates in $\operatorname{poly}(m,n)$ space and with $d^{O(n)}$ time complexity for $n$-qubit circuits, emphasizing scenarios where circuit depth $d$ is much smaller than the number of gates $m$.
3. Non-Relativizing Nature of Strong Supremacy Theorems:
   • The paper shows that any supremacy theorems proving $\mathsf{SampBPP} \neq \mathsf{SampBQP}$ and their consequences on polynomial hierarchy collapses must use non-relativizing techniques. This highlights the need for non-traditional complexity theoretical approaches in proving robust quantum supremacy claims.
4. Maximal Quantum Supremacy in Black-Box Models:
   • Through analysis of Fourier Sampling and Fourier Fishing problems, the authors demonstrate the maximal separation between randomized and quantum query complexities. They resolve conjectures regarding the extreme difficulty of simulating quantum behavior with classical resources in a black-box setting, marking a 1 versus linear query separation.
5. Quantum Supremacy Relative to Oracles in $\mathsf{P/poly}$:
   • Finally, the paper shows that under minimal assumptions, such as the existence of one-way functions, quantum supremacy can be established even relative to efficient oracles in $\mathsf{P/poly}$. This supports further exploration of feasible quantum supremacy experiments using computationally realistic oracles.

Theoretical and Practical Implications

By laying the complexity-theoretic foundation, the paper positions quantum supremacy not only as a technological challenge but also as a cornerstone for advancing theoretical computer science. The results here verbalize the hardness assumptions and computational boundaries critical in assessing quantum advantage for specific tasks.

The theoretical implications extend to refining the understanding of sampling problems – integral to quantum supremacy – and encourage non-relativizing approaches in proving complexity class separations. Practically, the research provides guidance for experimental setups, offering methodologies to validate claims of quantum advantage against classical simulation limits reliably.

Future Directions

This work opens several pathways for further research. The verification of QUATH, exploring new hardness conjectures that address proposals beyond circuit sampling, and elaborating the robustness of quantum supremacy under varied computational assumptions are primary areas. Additionally, another prospective domain is adjusting the error thresholds for sampling problems, allowing for broader adaptability in experimental tests. Investigating the optimality of the proposed algorithms and further clarifying the role of pseudo-randomness in realistic quantum computing environments will also contribute important insights going forward.

In summary, the paper pushes the boundaries of how quantum supremacy can be effectively demonstrated, merging complexity theory and quantum computing in innovative ways, thus setting a standard for both theoretical inquiry and experimental design.