Quantum leader election

Abstract

A group of n individuals $A_{1},\ldots A_{n}$ who do not trust each other and are located far away from each other, want to select a leader. This is the leader election problem, a natural extension of the coin flipping problem to n players. We want a protocol which will guarantee that an honest player will have at least $\frac{1}{n}-\epsilon$ chance of winning ($\forall \epsilon >0$), regardless of what the other players do (whether they are honest, cheating alone or in groups). It is known to be impossible classically. This work gives a simple algorithm that does it, based on the weak coin flipping protocol with arbitrarily small bias derived by Mochon (Quantum weak coin flipping with arbitrarily small bias, arXiv:0711.4114, 2000) in 2007, and recently published and simplified in Aharonov et al. (SIAM J Comput, 2016). A protocol with linear number of coin flipping rounds is quite simple to achieve; we further provide an improvement to logarithmic number of coin flipping rounds. This is a much improved journal version of a preprint posted in 2009; the first protocol with linear number of rounds was achieved independently also by Aharon and Silman (New J Phys 12:033027, 2010) around the same time.

Appendix: weak coin flipping

Let P be a weak coin flipping protocol, with $P_{B}^{*}$ the maximal cheating probability of Bob.

We want to run two instances of P, one after the other (not even at the same time).

We will define (see [3, 9] for full details) for P:

• Let $\mathcal {H=A\otimes M\otimes B}$ be the Hilbert space of the system.

• $\mid \psi _{0}>=\mid \psi _{A,0}>\mid \psi _{M,0}>\mid \psi _{B,0}>$ is the initial state of the system.

• Let there be n (even) stages, and i denote the current stage.

• On the odd stages i, Alice will apply a unitary $U_{A,i}$ on $\mathcal {A\otimes M}$.

• On the even stages, Bob will apply a unitary $U_{B,i}$ on $\mathcal {M\otimes B}$.

• Let $\mid \psi _{i}>$ be the state of the system in the $i_{th}$ stage.

• Let $\rho _{A,i}=Tr_{\mathcal {M\otimes B}}(\mid \psi _{i}> <\psi _{i}\mid )$ be the density matrix of Alice in the $i_{th}$ stage.

• Alice’s initial state (density matrix) is $^{(i)}\rho _{A,0}=|\psi _{A,0}> <\psi _{A,0}|$.

• For even state i, we have $^{(ii)}\rho _{A,i}=\rho _{A,i-1}$.

• Let ${\tilde{\rho }}_{A,i}$ be the state of $\mathcal {A\otimes M}$ after Alice gets the $i_{th}$ message.

• For odd i: $^{(iii)}\rho _{A,i}=Tr_{{\mathcal {M}}}({\tilde{\rho }}_{A,i})$, $^{(iv)}\rho _{A,i}=Tr_{{\mathcal {M}}}(U_{A,i}{\tilde{\rho }}_{A,i-1}U_{A,i}^{\dagger })$.

We know that regardless of Bob’s actions (see [3, 9] for full proof):

$$\begin{aligned} P_{B}^{*}\le \max Tr[\Pi _{A,1}\rho _{A,n}] \end{aligned}$$

(1)

where the maximization is done over all density matrices $\rho$ that satisfies the conditions (i)–(iv).