Impossibility of Classically Simulating One-Clean-Qubit Model with Multiplicative Error

The one-clean-qubit model (or the deterministic quantum computation with one quantum bit model) is a restricted model of quantum computing where all but a single input qubits are maximally mixed. It is known that the probability distribution of measurement results on three output qubits of the one-clean-qubit model cannot be classically efficiently sampled within a constant multiplicative error unless the polynomial-time hierarchy collapses to the third level [T. Morimae, K. Fujii, and J. F. Fitzsimons, Phys. Rev. Lett. 112, 130502 (2014)]. It was open whether we can keep the no-go result while reducing the number of output qubits from three to one. Here, we solve the open problem affirmatively. We also show that the third-level collapse of the polynomial-time hierarchy can be strengthened to the second-level one. The strengthening of the collapse level from the third to the second also holds for other subuniversal models such as the instantaneous quantum polynomial model [M. Bremner, R. Jozsa, and D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)] and the boson sampling model [S. Aaronson and A. Arkhipov, STOC 2011, p. 333]. We additionally study the classical simulatability of the one-clean-qubit model with further restrictions on the circuit depth or the gate types.

Figure 1

The $\mathrm{DQC}{1}_3$ circuit used to show $\text{postBQP}\subseteq {\text{postDQC}1}_3$ in Ref. [18]. The slash ($/$) means multiple qubits (in this case, $n-2$ qubits), and $X$ is applied on each qubit.

Figure 2

The $\mathrm{DQC}{1}_1$ circuit used to show $\mathrm{NQP}\subseteq \mathrm{NP}$.