Partial distinguishability as a coherence resource in boson sampling

Abstract

Quantum coherence is a useful resource that is consumed to accomplish several tasks that classical devices are hard to fulfill. Particularly, it is considered to be the origin of quantum speedup for many computational algorithms. In this work, we interpret the computational time cost of boson sampling with partially distinguishable photons from the perspective of coherence resource theory. With incoherent operations that preserve the diagonal elements of quantum states up to permutation, which we name permuted genuinely incoherent operation, we present some evidence that the decrease of coherence corresponds to a computationally less complex system of partially distinguishable boson sampling. Our result shows that coherence is one of crucial resources for the computational time cost of boson sampling. We expect our work presents an insight to understand the quantum complexity of the linear optical network system.

Appendices

The upper bound of transition probability with ${\text {perm}}(|{\mathcal {S}}|)$

A slight modification of Eq. (51) in Ref. [11] gives

$$\begin{aligned} P({\vec {n}},{\vec {m}})&= \Big |\sum _{\sigma \in S_N }{\text {perm}}(V\odot V^*_{\sigma , \mathbb {I}})\Big (\prod _i {\mathcal {S}}_{i, \sigma _i} \Big )\Big | \nonumber \\&\le {\text {perm}}(V\odot V^*)\sum _{\sigma \in S_N}\Big | \prod _i {\mathcal {S}}_{i,\sigma _i}\Big | \equiv P_\mathbb {I}[{\text {perm}}(|{\mathcal {S}}|)]. \end{aligned}$$

(15)

where the inequality comes from the relation $\big |{\text {perm}}(V\odot V^*_{\mathbb {I},\sigma } ) \big | \le {\text {perm}}(V\odot V^*)\equiv P_{\mathbb {I}}$ for any permutation $\sigma$. Using the monotonicity of ${\text {perm}}(|{\mathcal {S}}|)$ from Theorem 3, we can see that a pGIO on ${\mathcal {S}}$ decreases the upper bound of the transition probability $P({\vec {n}},{\vec {m}})$. The unitarity condition of V in Eq. (15) provides a more rigorous upper bound condition for the equation. Indeed, since $V\odot V^*$ is a unistochastic matrix (a doubly stochastic matrix whose elements are the absolute squares of the elements of a unitary matrix), the upper and lower bounds for $P_{\mathbb {I}} = {\text {perm}}(V\odot V^*)$ are given using the result in Ref. [50] by

$$\begin{aligned} F(V\odot V^*) \le {\text {perm}}(V\odot V^*) \le 2^NF(V\odot V^*) , \end{aligned}$$

(16)

where $F(V\odot V^*) \equiv \prod _{i,j=1}^N(1-|V_{ij}|^2)^{1-|V_{ij}|^2 }$. Hence, Eq. (15) can be rewritten as

$$\begin{aligned} P({\vec {n}},{\vec {m}}) \le 2^N F(V\odot V^*)[{\text {perm}}(|{\mathcal {S}}|)]. \end{aligned}$$

(17)

Alternative algorithm example

($N=4$) The runtime using Eq. (12) is $(2^{6}4^3)/2=2048$. However, when $S_{13}=S_{24}=S_{34}=0$, the summations with the following (R, S) become zero:

$$\begin{aligned} (R,S)=(\{1\}, \{3 \}),\quad (\{2\}, \{4 \}), (\{3\}, \{4 \}),\quad (\{3\}, \{1,4 \}), \quad (\{4\}, \{2,3 \}), \end{aligned}$$

(18)

which contains 4, 4, 8, and 8 terms, respectively. Therefore, the resulting runtime decreases to 2024.