Self-Testing of Universal and Fault-Tolerant Sets of Quantum Gates

Wim van Dam, Frédéric Magniez, Michele Mosca, Miklos Santha
2007 SIAM journal on computing (Print)  
We consider the design of self-testers for quantum gates. A self-tester for the gates F 1 , . . . , F m is a classical procedure which, given any gates G 1 , . . . , G m , decides with high probability if each G i is close to F i . This decision has to rely only on measuring in the computational basis the effect of iterating the gates on the classical states. It turns out that instead of individual gates, we can only design procedures for families of gates. To achieve our goal we borrow some
more » ... gant ideas of the theory of program testing: we characterize the gate families by specific properties, we develop a theory of robustness for them, and show that they lead to self-testers. In particular we prove that the universal and fault-tolerant set of gates consisting of a Hadamard gate, a c-NOT gate, and a phase rotation gate of angle π/4 is self-testable. the projection of the superposition to the subspace spanned by the basis states which are compatible with the outcome. As a result of a measurement, the state of the system becomes this projected state. The most convenient way to describe all possible operations on a quantum register is the formalism of 'density matrices'. In this approach, which differs from the Dirac notation, the quantum operations are described by completely positive superoperators (CPSOs) that act on matrices. Those density matrices describe mixed states (that is, a classical probability distributions over pure quantum states), and the CPSOs correspond exactly to all the physically allowed transformations on them. Such a model of quantum circuits with mixed states was described by Aharonov, Kitaev and Nisan[AKN98], and we will adopt it here. The unitary quantum gates of the standard model and measurements are special CPSOs. CPSOs can be simulated by unitary quantum gates on a larger number of qubits, and in [AKN98] it was shown that the computational powers of the two models are polynomially equivalent. Unitary quantum gates for small number of qubits were extensively studied. One reason is that although quantum gates for up to three qubits have already been built, constructing gates for large numbers seems to be elusive. Another reason is that universal sets of gates can be built from them, which means that they can simulate (approximately) any unitary transformation on an arbitrary number of qubits. The first universal quantum gate which operates on three qubits was identified by Deutsch [Deu89] . After a long sequence of works on universal quantum gates [DiV95, Bar95, DBE95, Llo95, BBC + 95, Sho96, KLZ96, Kit97], Boykin et al. [BMP + 99] have recently shown that the set consisting of a Hadamard gate, a c-NOT gate, and a phase rotation gate of angle π/4 is universal. In order to form a practical basis for quantum computation, a universal set must also be able to operate in a noisy environment, and therefore has to be fault-tolerant[Sho95, Sho96, AB97, Kit97, KLZ98]. The above set of three gates has the additional advantage of also being fault-tolerant. Experimental procedures for determining the properties of quantum "black boxes" were given by Chuang and Nielsen[CN97] and Poyatos, Cirac and Zoller[PCZ97], however these procedures implicitly require apparatus that has already been tested and characterized. The idea of self-testing in quantum devices is implicit in the work of Adleman, Demarrais and Huang[ADH97]. They have developed a procedure by which a quantum Turing machine is able to estimate its internal angle by its own means under the hypothesis that the machine is unitary. In the context of quantum cryptography Mayers and Yao[MY98] have designed tests for deciding if a photon source is perfect. These tests guarantee that if source passes them then it is adequate for the security of the Bennett-Brassard[BB84] quantum key distribution protocol. In this paper we develop the theory of self-testing of quantum gates by classical procedures. Given a CPSO G for n qubits, and a family F of unitary CPSOs, we would like to decide if G belongs to F. Intuitively, a self-tester is a classical program which should answer this question by interacting with the CPSOs to be tested only by observing classical outcomes of experiments that realize the CPSOs. More precisely, it will be a probabilistic algorithm which is able to access G as a black box in the following sense: it can prepare the classical states w ∈ {0, 1} n , can iterate G on them, and can make a measurement in the computational basis. The access must be seen as a whole, performed by a specific, experimental oracle for G: once the basis state w and the number of iterations k have been specified, the program in one step gets back one of the possible probabilistic outcomes of measuring the state of the system after G is iterated k-times on w. The intermediate quantum states of this process cannot be used by the program, which cannot perform any other quantum operations either. For 0 ≤ δ 1 ≤ δ 2 , such an algorithm will be a (δ 1 , δ 2 )-tester for F if for every CPSO G, whenever the distance of G and F is at most δ 1 (in some norm), it accepts with high probability, and whenever the same distance is greater than δ 2 , it rejects with high probability, where the probability is taken over the measurements performed by the oracle and by the internal coin tosses of
doi:10.1137/s0097539702404377 fatcat:taebrrlf7fhejcagg267hqt7de