We present a formalism that captures the process of proving quantum superiority to skeptics as an interactive game between two agents, supervised by a referee. Bob, is sampling from a classical distribution on a quantum device that is supposed to demonstrate a quantum advantage. The other player, the skeptical Alice, is then allowed to propose mock distributions supposed to reproduce Bob's device's statistics. He then needs to provide witness functions to prove that Alice's proposed mock distributions cannot properly approximate his device. Within this framework, we establish three results. First, for random quantum circuits, Bob being able to efficiently distinguish his distribution from Alice's implies efficient approximate simulation of the distribution. Secondly, finding a polynomial time function to distinguish the output of random circuits from the uniform distribution can also spoof the heavy output generation problem in polynomial time. This pinpoints that exponential resources may be unavoidable for even the most basic verification tasks in the setting of random quantum circuits. Beyond this setting, by employing strong data processing inequalities, our framework allows us to analyse the effect of noise on classical simulability and verification of more general near-term quantum advantage proposals.
The easiest way to establish quantum advantage would be to solve a well-established hard computational problem, such as factoring large numbers or simulating large-sized molecules. Unfortunately, although well-known quantum algorithms provide speedups for these problems, their implementation is likely beyond the power of the devices that will be available in the following years.
Thus, the community focused on quantum advantage proposals based on sampling from the outcomes of random quantum circuits. This is because current quantum devices can sample from (noisy) circuits, and there is strong complexity-theoretic evidence that this is a challenging task for classical computers.
Unfortunately, this random circuit sampling is not known to have practical applications. Furthermore, it is not known how to certify that the quantum device is indeed sampling from a distribution close to the target one in some metric without employing exponential classical computational time. In fact, it is not even known how to efficiently distinguish the output of a random quantum circuit from a fair coin toss.
In this work, we show that the lack of efficient ways to distinguish the outputs of quantum circuits is intimately related to the hardness of their simulation. We exploit a framework where most of the existing approaches to certify quantum advantage can be understood as a game between an agent that wishes to convince the community to have reached quantum advantage (Bob), and a skeptical member (Alice).
In this game, Alice is allowed to propose an alternative hypothesis to what Bob's device is doing, say just sampling from fair coins. It is then Bob's job to propose an (efficient) test refuting Alice's hypothesis by pointing out that Alice cannot reproduce specific statistics of his distribution. Alice and Bob then play an interactive game of new proposal and refutation test proposals until one of the two players can not propose a new distribution (Alice) or a novel test (Bob) and concedes defeat.
Our main result is that Bob can never win this game in the setting of random quantum circuits using efficiently computable test functions. The reason is that the existence of an efficient way of distinguishing his distributions from Alice's would also allow Alice to simulate Bob's device efficiently. As it is not believed that the outputs of random quantum circuits can be simulated efficiently classically, our results indicate that for such problems, efficient verification strategies are not possible. In addition, we show that even the existence of an efficient test that distinguishes the output from perfectly random coins seems unlikely, as it is in direct contradiction with a recent complexity theory conjecture.
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