Simulating the quantum switch with quantum circuits is computationally hard

TL;DR

This work establishes an exponential quantum query complexity separation between indefinite causal order, exemplified by the quantum switch $\\mathcal{S}$, and quantum circuits with fixed or classically-controlled causal order. It proves that deterministic exact simulation of $\\mathcal{S}$ on all $n$-qubit channels by any circuit using $k_A$ calls to $A$ and $k_B$ calls to $B$ is impossible whenever $k_A \leq \max(2, 2^{n}-1)$ with $k_B=1$, and that even probabilistic or approximate simulations fail for several small $(k_A,k_B)$ pairs; an SDP framework is developed to bound the maximal simulation success probability $p$. Conversely, a go-theorem shows that if $A$ is a bipartite unitary and $B$ is general, there exists a circuit with $(k_A,k_B)=(2,1)$ achieving $\\mathcal{S}\\otimes\\mathcal{I}(A,B)$, illustrating a nuanced boundary between universality and case-specific simulability. The results are supported by computer-assisted SDP proofs and detailed discussion of restricted-simulation scenarios, with implications for interpreting quantum-switch experiments and for the theory of higher-order quantum computation. The work thus suggests that deterministic, universal simulation of indefinite causal order may require exponentially many queries, reinforcing the computational distinctiveness of processes with indefinite causal order.

Abstract

Higher-order transformations acting on input quantum channels in an indefinite causal order, such as the quantum switch, cannot be described by quantum circuits using the same number of calls to the input channels. A natural question is whether they can be simulated, i.e., whether their action can be exactly and deterministically reproduced by a quantum circuit with more calls to the input channels. Here, we prove that the quantum switch acting on two $n$-qubit channels cannot be simulated by any quantum circuit using $k$ calls to one channel and one to the other, if $k<2^n$. This establishes an exponential separation in quantum query complexity between processes with indefinite causal order and quantum circuits. Moreover, even with one extra call to both input channels, such a simulation remains impossible. We further demonstrate the robustness of this separation by extending the result to probabilistic and approximate simulations scenarios.

Figures (9)

• Figure 1: The quantum switch transformation. The quantum switch $\mathcal{S}$ is a higher-order transformation that takes as input any two quantum channels $A$ and $B$ and transforms them into a another quantum channel $\mathcal{S}(A,B)$. The resulting channel $\mathcal{S}(A,B)$ acts on a qubit control system and a qudit target system.
• Figure 2: Simulating the quantum switch. A higher-order transformation $\mathcal{C}$ corresponding to an open-slot quantum circuit with fixed or classically-controlled causal order, which acts on several copies of the input quantum channels $A$ and $B$, is a simulation of the quantum switch $\mathcal{S}$ if it reproduces the action of the quantum switch on all arbitrary channels $A$ and $B$.
• Figure 3: The extended quantum switch transformation. The quantum switch $\mathcal{S}$ is a higher-order transformation that takes as input any two quantum channels $A$ and $B$ and transforms them into another quantum channel, $\mathcal{S}\otimes\mathcal{I}(A,B)$, even when acting only on part of these channels.
• Figure 4: Robustness of the impossibility of simulation. Maximum probability of success $p$ of simulating any higher-order transformation that is $\epsilon$-close to the quantum switch using quantum combs of different orders or QC-CCs, as a function of $\epsilon$. In plot $(a)$, we show the case where $k_A=k_B=1$. Note how the QC-CC simulation curve numerically coincides with the comb simulation curve. In plot $(b)$, we show the case where $k_A=2$ and $k_B=1$. In this case, for some values of $\epsilon$, QC-CCs show an advantage with respect to quantum combs.
• Figure B1: General simulation of the quantum switch. A higher-order transformation $\mathcal{C}$, which can be a quantum comb or a QC-CC, that acts on part of several copies of input bipartite quantum channels $A$ and $B$ is a simulation of the quantum switch $\mathcal{S}$ if it reproduces the action of the quantum switch on all arbitrary pairs of bipartite channels $A$ and $B$.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Conjecture 1
• Theorem 1
• proof
• Theorem 2: expanded
• proof
• Lemma 1
• proof