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IQP Circuits, Different Constructions and Noise Effects

Longxiang Yuan

1 Background

Instantaneous Quantum Polynomial (IQP) verification holds a pivotal role in demonstrating quantum computational advantage due to its efficient classical verification and inherent resistance to classical simulation.

There are various different important families of IQP circuits. First, a central construction is the X-program circuits with \(\theta = \pi/8\) [SB09, BCJ23]. Circuits in this family has the form \[\begin{equation} U_{H} = \prod_{p \in \mathrm{row}(H)} \exp \Bigl(\frac{i\pi}{8} X^{p} \Bigr) \end{equation}\] where \(H \in \mathbb{F}_2^{m \times n}\) and \(X^{p} := X^{p_1} \otimes X^{p_2} \otimes \cdots \otimes X^{p_n}\).

Second, we examine circuits constructed with an initial layer of Hadamard gates applied to all qubits, followed by a diagonal component comprising T, CS, and CCZ gates, and concluding with a final layer of Hadamard gates on all qubits. In this work, we refer to this class of circuits as IQP-3 circuits.

In IQP verification protocols, the output state of an IQP circuit initialized in the all-zero state is measured, yielding an n-bit string \(x\). Verification proceeds by computing a linear statistic called the signal, defined as \(s \cdot x = \sum_{i=1}^n s_i x_i \pmod{2}\), where \(s\) is a predetermined \(n\)-bit string. Since the signal is a random variable that varies with each measurement of \(x\), the procedure is repeated multiple times to estimate its expected value.

2 Problems

2.1 Relationship between X-Program and IQP-3 Circuits

Are X-program circuits with \(\theta = \pi/8\) and IQP-3 circuits equivalent? Specifically:

  1. Can every X-program circuit be recompiled into an equivalent IQP-3 circuit (up to a global phase)?
  2. Conversely, does every IQP-3 circuit admit an X-program representation?

2.2 Analyze the noise effect for measuring \(s \cdot x\)

When an IQP circuit is executed on noisy quantum hardware, gate-level errors accumulate during computation. These errors affect the measurement statistics when estimating the expectation value of the output signal \(s \cdot x\). What are the noise rate constraints based on circuit parameters like gate count and depth to ensure reliable signal measurement?

3 Contacts

Longxiang Yuan (ylx23@mails.tsinghua.edu.cn)

Zhengfeng Ji (jizhengfeng@mail.tsinghua.edu.cn)

4 References

[SB09] D. Shepherd and M. J. Bremner. Temporally unstructured quantum computation, Proc. R. Soc. A 465, 1413 (2009).

[BCJ23] Michael J. Bremner, Bin Cheng, and Zhengfeng Ji. IQP Sampling and Verifiable Quantum Advantage: Stabilizer Scheme and Classical Security, August 2023. arXiv:2308.07152 [quant-ph].

[GH23] David Gross and Dominik Hangleiter. Secret extraction attacks against obfuscated IQP circuits, December 2023. arXiv:2312.10156 [quant-ph].