Table 2 Estimates of the input sizes n required to demonstrate quantum advantage using constant-depth quantum circuits in both noise-free and noise-affected settings, based on the quantum upper and classical lower bounds determined in Section II B, Section II C, and Section II D

From: Unconditional advantage of noisy qudit quantum circuits over biased threshold circuits in constant depth

 

\({{\mathsf{NC}}}^{0}\) regime

\({{\mathsf{bPTC}}}^{0}(k)\) regime

Quantum circuit type

\({\mathsf{Min}}\,{F}^{*}\)

F = 2

⋯  

F = 8

k = 1

k = 2

⋯  

k = n1/(5d)

Qubits

2540

9364

⋯  

1.6 × 108

4.3 × 1013

5.5 × 1016

⋯  

5.0 × 1026

Noisy qubits

-

7.0 × 1010

⋯  

1.5 × 1023

1.8 × 1038

8.0 × 1044

⋯  

1.1 × 1075

Qutrits

125162

1952660

⋯  

1.3 × 1010

5.5 × 1014

1.1 × 1018

⋯  

2.3 × 1030

Noisy qutrits

-

3.5 × 1011

⋯  

9.0 × 1030

3.5 × 1052

9.0 × 1060

⋯  

3.4 × 10101

Ququints

1.0 × 108

1.5 × 109

⋯  

8.0 × 1012

9.5 × 1014

1.7 × 1018

⋯  

5.0 × 1030

Noisy ququints

-

3.0 × 1014

⋯  

7.0 × 1033

5.5 × 1052

1.4 × 1061

⋯  

7.0 × 0101

  1. For the noise-free case, we consider the depth of four quantum circuits solving the \({{{{\mathcal{R}}}}}_{p}\) problems. These circuits, featuring 2n gates and all-to-all connectivity, generate the shortest possible solution strings, creating harder instances for classical circuits to replicate. The classical lower bounds for each problem are obtained from and depend on the deviation between the optimal classical and quantum winning strategies for the XOR non-local games \({{{{\mathcal{G}}}}}_{p}\). We also examine the minimal fan-in (F*) scenario by comparing quantum circuits with classical circuits of equal locality—that is, having the same fan-in as the quantum circuits in each layer. This yields the lowest resource estimates for direct comparisons. Additionally, we analyze exact-case hardness bounds for the qubit setting to establish lower resource estimates. For the remaining qudit dimensions, we rely on average-case hardness to derive comparable estimates.
  2. In the noisy setting, we analyze depth-9 error-corrected quantum circuits for qubits and depth-11 circuits for qudits. The latter qudit circuits include additional logical operations implemented in a noise-resilient manner, requiring noise-tolerant versions based on the qudit surface code. We assume a code distance of order \(\log (n)\), as no specific error threshold is defined for the local stochastic noise. This threshold is left as a parameter for further investigation and potential alignment with quantum hardware advancements.
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