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Complexity and hardness of random peaked circuits

Yuxuan Zhang

TL;DR

This work analyzes random peaked circuits (RPCs) as a candidate for verifiable quantum advantage by constructing peaked unitaries via postselection on a designated output string. The authors prove that RPCs are incompressible and require $\tilde{\Omega}(nk)$ gates on average, using unitary $k$-design properties and a packing-based argument. They establish hardness results for simulating RPCs, including #P-hardness for exact amplitudes and PromiseBQP-completeness for additive errors, and provide worst-to-average reductions. The paper also develops a practical verifiable protocol by leveraging variational search and circuit stitching to generate large RPCs, and demonstrates robustness to sparse classical noise, plus de-biasing under depolarizing noise. Collectively, these results position RPCs as a scalable, verifiable path to near-term quantum advantage with rigorous complexity-theoretic guarantees and practical verification strategies.

Abstract

Near-term feasibility, classical hardness, and verifiability are the three requirements for demonstrating quantum advantage; most existing quantum advantage proposals achieve at most two. A promising candidate recently proposed is through randomly generated peaked circuits. In this work, we study an explicit construction for random peaked circuits: first selecting a random circuit $C$ of polynomial size, which forms a $k$-design. Subsequently, a second random circuit $C'$ is chosen from the same architecture, subject to a postselection criterion: $C'$ must exhibit a high overlap with $C$ in one of their rows. Utilizing unitary design properties, we demonstrate that the circuits generated by this method are non-trivial; specifically, $C'$ is provably far from $C^\dagger$. Indeed, with overwhelmingly high probability, a random peaked circuit generated this way is non-compressible and is of circuit complexity $\tilde Ω(nk)$. This resolves an open problem posed by Aaronson in 2022. Secondly, we analytically establish that estimating the peakedness of a random peaked circuit to within a $2^{-\text{poly}(n)}$ additive error, is average-case \#P-hard. When the additive error is relaxed to $1/\text{poly}(n)$, we note that the worst-case scenario for this problem is BQP-complete. Under widely accepted assumptions on random quantum circuits, we identify a regime where no classical polynomial-time sequential simulator attains inverse-polynomial additive accuracy on the peak on a non-negligible fraction of instances. Thirdly, we study using peaked circuits as a practical attempt for a verifiable quantum advantage protocol. While the postselection method for generating peaked circuits could be costly, we demonstrate that numerical search for $C'$ with randomized initialization successfully returns a random peaked circuit, achieving the properties as theoretically predicted.

Complexity and hardness of random peaked circuits

TL;DR

This work analyzes random peaked circuits (RPCs) as a candidate for verifiable quantum advantage by constructing peaked unitaries via postselection on a designated output string. The authors prove that RPCs are incompressible and require gates on average, using unitary -design properties and a packing-based argument. They establish hardness results for simulating RPCs, including #P-hardness for exact amplitudes and PromiseBQP-completeness for additive errors, and provide worst-to-average reductions. The paper also develops a practical verifiable protocol by leveraging variational search and circuit stitching to generate large RPCs, and demonstrates robustness to sparse classical noise, plus de-biasing under depolarizing noise. Collectively, these results position RPCs as a scalable, verifiable path to near-term quantum advantage with rigorous complexity-theoretic guarantees and practical verification strategies.

Abstract

Near-term feasibility, classical hardness, and verifiability are the three requirements for demonstrating quantum advantage; most existing quantum advantage proposals achieve at most two. A promising candidate recently proposed is through randomly generated peaked circuits. In this work, we study an explicit construction for random peaked circuits: first selecting a random circuit of polynomial size, which forms a -design. Subsequently, a second random circuit is chosen from the same architecture, subject to a postselection criterion: must exhibit a high overlap with in one of their rows. Utilizing unitary design properties, we demonstrate that the circuits generated by this method are non-trivial; specifically, is provably far from . Indeed, with overwhelmingly high probability, a random peaked circuit generated this way is non-compressible and is of circuit complexity . This resolves an open problem posed by Aaronson in 2022. Secondly, we analytically establish that estimating the peakedness of a random peaked circuit to within a additive error, is average-case \#P-hard. When the additive error is relaxed to , we note that the worst-case scenario for this problem is BQP-complete. Under widely accepted assumptions on random quantum circuits, we identify a regime where no classical polynomial-time sequential simulator attains inverse-polynomial additive accuracy on the peak on a non-negligible fraction of instances. Thirdly, we study using peaked circuits as a practical attempt for a verifiable quantum advantage protocol. While the postselection method for generating peaked circuits could be costly, we demonstrate that numerical search for with randomized initialization successfully returns a random peaked circuit, achieving the properties as theoretically predicted.

Paper Structure

This paper contains 33 sections, 35 theorems, 117 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Circuit complexity of random peaked circuits
  3. Constructing RPC's with postselection
  4. Block decomposition under first-column conditioning
  5. Packing-based circuit lower bounds for the peaked ensemble
  6. Hardness of simulating peaked circuits
  7. 'Almost exact' simulation is #P Hard
  8. Worst case hardness via a peaked embedding
  9. Worst-to-average reduction with a polynomial method
  10. Additive inverse-poly error is PromiseBQP-complete
  11. (i) MPS contraction.
  12. (ii) Peaked shallow circuit simulation.
  13. Compiler as an obfuscator
  14. Verifiable quantum advantage with RPC sampling
  15. Bounding the postselection success probability
  16. ...and 18 more sections

Key Result

Theorem 1.3

Suppose we have some architecture with poly-sized random gates that forms a unitary $k$-design. The obtained $C'^\dagger C$ according to def: pcc at least requires $\tilde{\Omega}(nk)$ gates to implement with overwhelmingly high probability.

Figures (2)

  • Figure 1: Overview of the results in this paper: We consider generating peaked circuits from random circuits and postselectiing on the output distribution being peaked. We analytically prove that the peaked circuits generated this way are incompressible. Furthermore, we show that estimating the output weight of the peaked string can be average case computationally hard for a classical computer, whereas a quantum computer can obtain the peakedness by simply running the circuit and perform sampling. Compared to conventional random circuit sampling protocol, peaked circuits are easy to verify: the peaked string and its corresponding weight serves as a witness for efficient verification.
  • Figure 2: Left: In practice, generating RPC from postselection has an extremely small success probability. Therefore, we randomly generate $C$ and variationally optimize $C'$ with randomized initializations and gradient-based optimizer, Adam. This figure reflects an artist's impression of the process: Adam kingma2014adam, the tireless optimizer start with a random location in the parameter space. Following the local gradient in the landscape, Adam find the closest peaked circuit to the initialization. This search process is a practical rescue for generating random peaked circuits. Middle: To construct larger peaked circuits that goes beyond the size of classical simulability, we show that it is possible to combine small peaked circuits while still reliably keep tracking of the peaked string and peakedness. Right: Through a comparision in the Hilbert-Schmidt overlap between $C$ and $C'$, the plot shows that the circuit generated with Adam optimization has similar properties to those generated by postselection (as predicted by our theory). The numerical result is averaged over 100 random instances of $C$. This clearly shows that, on average, the random peaked circuits generated numerically are far from identity and very obfuscated.

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2: Random Peaked Circuit (RPC) Construction via Postselection.
  • Theorem 1.3: Circuit complexity of random peaked circuits. Informal.
  • Theorem 1.4: Hardness of simulating random peaked circuits. Informal.
  • Theorem 2.1: Exact block--Haar decomposition
  • proof
  • Corollary 2.2: Peaked ensemble under Haar sampling
  • Proposition 2.3: Block--design decomposition up to degree-$k$ moments
  • proof : Proof sketch
  • Corollary 2.4: Peaked ensemble under $k$-design sampling
  • ...and 56 more