Continuous-Variable Quantum Computing and its Applications to Cryptography

Abstract

We propose a quantum cryptography based on an algorithm for determining a function using continuous-variable entangled states. The security of our cryptography is based on the Ekert 1991 protocol, which uses an entangled state. Eavesdropping destroys the entangled state. Alice selects a secret function from the very large number of possible function types. Bob’s aim is to determine the selected function (a key) without an eavesdropper learning it. In order for both Alice and Bob to be able to select the same function classically, in the worst case Bob requires a very large number of queries to Alice. In the quantum case however, Bob requires just a single query. By measuring the single entangled state, which is sent to him by Alice, Bob can obtain the function that Alice has selected. This quantum key distribution method is faster than the very large number of classical queries that would be required in the classical case.

Appendix: The Phase Kick-Back Formation

We have the following formula by the phase kick-back formation [18]:

$$ \begin{array}{@{}rcl@{}} U_{f}|x\rangle|\phi_{d}\rangle =\omega^{f(x)}|x\rangle|\phi_{d}\rangle. \end{array} $$

(1)

In what follows, we discuss the rationale behind the above relation (1). Consider the action of the U_f gate on the state |x〉|ϕ_d〉. Each summand in |ϕ_d〉 is of the form ω^d−j|j〉. We observe that

$$ \begin{array}{@{}rcl@{}} U_{f}\omega^{d-j}|x\rangle|j\rangle =\omega^{d-j} |x\rangle|(j+f(x))\ \text{mod}\ d\rangle. \end{array} $$

(2)

A variable k is introduced such that f(x) + j = k, from which it follows that d − j = d + f(x) − k. Thus, (2) becomes

$$ \begin{array}{@{}rcl@{}} U_{f}\omega^{d-j}|x\rangle|j\rangle =\omega^{f(x)} \omega^{d-k}|x\rangle|k\ \text{mod}\ d\rangle. \end{array} $$

(3)

If k < d we have that |k mod d〉 = |k〉 and thus the summands in |ϕ_d〉 for which k < d are transformed as follows:

$$ \begin{array}{@{}rcl@{}} U_{f}\omega^{d-j}|x\rangle|j\rangle =\omega^{f(x)}\omega^{d-k}|x\rangle|k\rangle. \end{array} $$

(4)

On the other hand, as both f(x) and j are bounded from above by d, k is strictly less than 2d. Thus, when d ≤ k < 2d, we have |k mod d〉 = |k − d〉. Let k − d = m. We have

$$ \begin{array}{@{}rcl@{}} &&\omega^{f(x)}\omega^{d-k}|x\rangle|k\ \text{mod}\ d\rangle =\omega^{f(x)}\omega^{-m}|x\rangle|m\rangle\\ &&=\omega^{f(x)}\omega^{d-m}|x\rangle|m\rangle. \end{array} $$

(5)

Hence, the summands in |ϕ_d〉 for which k ≥ d are transformed as follows:

$$ \begin{array}{@{}rcl@{}} U_{f}\omega^{d-j}|x\rangle|j\rangle =\omega^{f(x)}\omega^{d-m}|x\rangle|m\rangle. \end{array} $$

(6)

Finally, regarding (4) and (6), we have

$$ \begin{array}{@{}rcl@{}} U_{f}|x\rangle|\phi_{d}\rangle=\omega^{f(x)}|x\rangle|\phi_{d}\rangle. \end{array} $$

(7)

Therefore, the relation (1) holds.