Efficient quantum voting with information-theoretic security

1 . A method of conducting a vote using a quantum network, the method comprising:

by a tally quantum node, generating a first plurality of ballot quantum states encoding a first bit string, each of the first plurality of ballot quantum states comprising a plurality of qubits;

distributing each of the first plurality of ballot quantum states to exactly one of a plurality of voter quantum nodes via a quantum network;

by at least one of the plurality of voter quantum nodes:

performing a projective measurement of the respective one of the first plurality of ballot quantum states for each voter quantum node of the at least one of the plurality of voter quantum nodes, and thereby determining a parity of a random pair of bits of the first bit string;

reading a vote;

computing a first encoded vote based on the parity and the vote; and

broadcasting the first encoded vote and an identifier of the pair of bits to each other of the plurality of voter quantum nodes; and

providing the first bit string to decode the first encoded vote.

2 . The method of claim 1 , further comprising:

based on the first bit string and the identifier, decoding the first encoded vote to determine a first decoded vote; and

tallying the first decoded vote.

3 . The method of claim 2 , further comprising outputting a result to a user according to the tally.

4 . The method of claim 1 , wherein each of the plurality of voter quantum nodes:

performs a projective measurement of the respective one of the first plurality of ballot quantum states for each voter quantum node of the plurality of voter quantum nodes, and thereby determines a parity of a random pair of bits of the first bit string;

receives a vote;

computes a first encoded vote based on the parity and the vote;

broadcasts the first encoded vote and an identifier of the pair of bits to each other of the plurality of voter quantum nodes.

5 . The method of claim 1 , further comprising:

generating a second plurality of ballot quantum states encoding a second bit string, each of the second plurality of ballot quantum states comprising a plurality of qubits;

distributing each of the second plurality of ballot quantum states to exactly one of the plurality of voter quantum nodes via the quantum network;

by at least one of the plurality of voter quantum nodes:

performing a projective measurement of the respective one of the second plurality of ballot quantum states for each voter quantum node of the at least one of the plurality of voter quantum nodes, and thereby determining a parity of a random pair of bits of the second bit string;

computing a second encoded vote based on the parity and the vote;

broadcasting the second encoded vote and an identifier of the pair of bits to each other of the plurality of voter quantum nodes;

providing the second bit string to decode the second encoded vote.

6 . The method of claim 5 , further comprising:

averaging the first and second encoded votes to determine the vote.

7 . The method of claim 1 , wherein the each of the first plurality of ballot quantum states is given by:

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wherein the first bit string has n bits and the x i denotes the i th bit in the first bit string.

8 . The method of claim 1 , wherein the quantum network is an optical fiber network.

9 . The method of claim 1 , wherein the plurality of qubits of each of the first plurality of ballot quantum states are photons.

10 . The method of claim 1 , further comprising:

generating the first bit string randomly.

11 . The method of claim 1 , further comprising:

preparing a shared Greenberger-Horne-Zeilinger state among the tally quantum node and the plurality of voter quantum nodes.

12 . The method of claim 11 , wherein preparing the shared Greenberger-Horne-Zeilinger state comprises:

distributing a ring of Bell pairs connecting the tally quantum node and the plurality of voter quantum nodes in a ring.

13 . The method of claim 11 , wherein distributing each of the first plurality of ballot quantum states comprises:

from the shared Greenberger-Horne-Zeilinger state, distilling at least one Bell pair between the tally quantum node and each of the plurality of voter quantum nodes.

14 . The method of claim 11 , wherein the first encoded vote and the identifier of the pair of bits are broadcast via the shared Greenberger-Horne-Zeilinger state.

15 . The method of claim 11 , further comprising:

the first encoded vote and the identifier of the pair of bits are broadcast via the shared Greenberger-Horne-Zeilinger state.

16 . The method of claim 11 , further comprising:

broadcasting by at least one of the plurality of quantum nodes a complaint bit via the GHZ state;

tallying the complaint bit by computing a multiparty secure sum on the GHZ state.

17 . The method of claim 1 , wherein computing the first encoded vote comprises adding the parity and the vote modulo 2.

18 . The method of claim 1 , further comprising:

forming an anonymous queue of the plurality of the plurality of voter quantum nodes,

wherein the first plurality of ballot quantum states are distributed in order of the anonymous queue.

19 . The method of claim 18 , wherein forming the anonymous queue comprises:

preparing a shared Greenberger-Horne-Zeilinger state among the tally quantum node and the plurality of voter quantum nodes;

each of the plurality of voter quantum nodes broadcasting a queue entry request via the shared Greenberger-Horne-Zeilinger state;

each of the plurality of voter quantum nodes detecting a successful queue entry request by computing a multiparty secure sum.

20 . The method of claim 18 , wherein the first encoded vote is broadcast in queue order.

21 . The method of claim 1 , wherein reading the vote comprises receiving a user input.

22 . A method of casting a vote using a quantum network, the method comprising:

receiving, at a first voter quantum node of a plurality of voter quantum nodes, a ballot quantum state via a quantum network, the ballot quantum state encoding a first bit string in a plurality of qubits;

performing a projective measurement of the ballot quantum state, and thereby determining a parity of a random pair of bits of the first bit string;

reading a vote;

computing a first encoded vote based on the parity and the vote; and

broadcasting, by the first voter quantum node, the first encoded vote and an identifier of the pair of bits to each other of the plurality of voter quantum nodes.

23 . A method of tallying a vote using a quantum network, the method comprising:

receiving a plurality of encoded votes, each with an associated identifier, wherein

each associated identifier identifies a pair of bits randomly selected from a predetermined bit string, and

each encoded vote is based on a vote and a parity, the parity being based on a projective measurement of a ballot quantum state encoding the predetermined bit string;

receiving the predetermined bit string;

based on the predetermined bit string and the associated identifiers, decoding the plurality of encoded votes; and

tallying the plurality of decoded votes.

24 . A system comprising:

a quantum network;

a tally quantum node operatively coupled to the quantum network; and

a plurality of voter quantum nodes operatively coupled to the quantum network, wherein:

the tally quantum node is configured to:

generate a first plurality of ballot quantum states encoding a first bit string, each of the first plurality of ballot quantum states comprising a plurality of qubits; and

distribute each of the first plurality of ballot quantum states to exactly one of the plurality of voter quantum nodes via the quantum network; and

each of the plurality of voter quantum nodes is configured to:

perform a projective measurement of the respective one of the first plurality of ballot quantum states for that voter quantum node, and thereby determining a parity of a random pair of bits of the first bit string;

read a vote;

compute a first encoded vote based on the parity and the vote; and

broadcast the first encoded vote and an identifier of the pair of bits to each other of the plurality of voter quantum nodes via the quantum network.