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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.
Problems
Relationship between X-Program
and IQP-3 Circuits
Are X-program circuits with \(\theta =
\pi/8\) and IQP-3 circuits equivalent? Specifically:
- Can every X-program circuit be recompiled into an equivalent IQP-3
circuit (up to a global phase)?
- Conversely, does every IQP-3 circuit admit an X-program
representation?
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?
Longxiang Yuan (ylx23@mails.tsinghua.edu.cn)
Zhengfeng Ji (jizhengfeng@mail.tsinghua.edu.cn)
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].