Magic and communication complexity

TL;DR

This work establishes a tight link between the amount of magic in quantum circuits and communication complexity, showing that low magic yields low communication across several models and that even adaptive, mid-circuit protocols inherit this efficiency. It introduces a suite of lower bounds that tie SMP and two-way communication costs to the number and structure of magic gates (including $T$-gates) in Clifford+Magic circuits, and uses these to prove Omega(n) magic-cost bounds for Toffoli_n and the quantum multiplexer. A general transformation is developed to convert $ abla Q\|^*$ protocols into private, classical $ abla R\|^*$ protocols with polynomial overhead when the referee’s action has constant $T$-depth, yielding PS M^* privacy and enabling exponential separations between $ abla R\|^*$ and $ abla R$ for partial Boolean functions. The results illuminate how quantum circuit depth and non-Clifford cost constrain classical communication, with implications for quantum-speedups in communication tasks and for nonlocal quantum computation strategies.

Abstract

We establish novel connections between magic in quantum circuits and communication complexity. In particular, we show that functions computable with low magic have low communication cost. Our first result shows that the $\mathsf{D}\|$ (deterministic simultaneous message passing) cost of a Boolean function $f$ is at most the number of single-qubit magic gates in a quantum circuit computing $f$ with any quantum advice state. If we allow mid-circuit measurements and adaptive circuits, we obtain an upper bound on the two-way communication complexity of $f$ in terms of the magic + measurement cost of the circuit for $f$. As an application, we obtain magic-count lower bounds of $Ω(n)$ for the $n$-qubit generalized Toffoli gate as well as the $n$-qubit quantum multiplexer. Our second result gives a general method to transform $\mathsf{Q}\|^*$ protocols (simultaneous quantum messages with shared entanglement) into $\mathsf{R}\|^*$ protocols (simultaneous classical messages with shared entanglement) which incurs only a polynomial blowup in the communication and entanglement complexity, provided the referee's action in the $\mathsf{Q}\|^*$ protocol is implementable in constant $T$-depth. The resulting $\mathsf{R}\|^*$ protocols satisfy strong privacy constraints and are $\mathsf{PSM}^*$ protocols (private simultaneous message passing with shared entanglement), where the referee learns almost nothing about the inputs other than the function value. As an application, we demonstrate $n$-bit partial Boolean functions whose $\mathsf{R}\|^*$ complexity is $\mathrm{polylog}(n)$ and whose $\mathsf{R}$ (interactive randomized) complexity is $n^{Ω(1)}$, establishing the first exponential separations between $\mathsf{R}\|^*$ and $\mathsf{R}$ for Boolean functions.

Figures (6)

• Figure 1: a) The simultaneous message passing ($\mathsf{D}\|$) setting. Alice receives input $x\in\{0,1\}^n$, Bob receives input $y\in\{0,1\}^n$, and the referee should output $f(x,y)$. Alice and Bob can not communicate with one another, but can each send a message to the referee. The $\mathsf{D}\|$ cost is the minimal number of bits of communication Alice and Bob must send. b) The $\mathsf{PSM}^*$ model, which has the same communication pattern as $\mathsf{D}\|$. In $\mathsf{PSM}^*$, Alice and Bob may share entanglement, which we indicate with the lower curved wire. We restrict the communication to be classical, which is indicated by the double-lined wires. Further, the messages are required to be private, meaning that the referee should learn $f(x,y)$ but no other information about $(x,y)$.
• Figure 2: Quantum versus Classical Communication. Here, an arrow from $A$ to $B$ denotes that $A$ exponentially outperforms $B$ for some task, with solid lines denoting functional tasks and dashed lines denoting relational ones. We use $2$ to denote interactive protocols, $1$ to denote one-way protocols and $\|$ to denote simultaneous protocols.
• Figure 3: (a) Circuit diagram showing the local implementation of a channel $\mathbfcal{N}$. (b) Circuit diagram showing the form of a non-local quantum computation. $\mathbfcal{V}^L$, $\mathbfcal{V}^R$, $\mathbfcal{W}^L$, and $\mathbfcal{W}^R$ are quantum channels. The goal is to simulate the local channel $\mathbfcal{N}$.
• Figure 4: Quantum circuit with $k$ magic gates that computes $f(x,y)$.
• Figure 5: The circuit simulated by the referee in our $\mathsf{D}\|$ protocol. The construction begins with a circuit (\ref{['fig:layeredcircuit']}) that computes $f(x,y)$ from inputs $(x,y)$ along with an advice state. Here, we run the circuit on the all-zeroes input, and Pauli corrections $\sigma_i$ are made just before each magic gate $M_i$ as necessary. One additional Pauli correction is made before the measurement. Each Pauli correction can be computed in the $\mathsf{D}\|$ model with constant communication, so the total communication cost is a constant times the number of magic gates.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Theorem 7
• Remark 8
• Theorem 9
• Theorem 10