In a recent letter [Information Processing Letters~104 (2007) 152-158], it has shown some sufficient conditions for commutativity of quantum weakest preconditions. This paper provides some alternative and simple characterizations for the commutativity of quantum weakest preconditions, i.e., Theorem 3.1, Theorem 3.2 and Proposition 3.3 in what follows. We also show that to characterize the commutativity of quantum weakest preconditions in terms of $[M,N]$ ($=MN-NM$) is hard in the sense of Proposition 4.1 and Proposition 4.2.
Deep Dive into Simple characterizations for commutativity of quantum weakest preconditions.
In a recent letter [Information Processing Letters~104 (2007) 152-158], it has shown some sufficient conditions for commutativity of quantum weakest preconditions. This paper provides some alternative and simple characterizations for the commutativity of quantum weakest preconditions, i.e., Theorem 3.1, Theorem 3.2 and Proposition 3.3 in what follows. We also show that to characterize the commutativity of quantum weakest preconditions in terms of $[M,N]$ ($=MN-NM$) is hard in the sense of Proposition 4.1 and Proposition 4.2.
The theory of quantum computation, including the subfield of semantics for quantum programming languages [1,2,3,4,5,6,7], develops rapidly. This is, to a large extent, owing to the motivation of Shor's quantum factoring algorithms [9] and Grover's searching algorithm [10].
Quantum algorithms is a very important research direction [11]. However, quantum algorithm currently are expressed at the very low level of quantum circuits [11] which is a disadvantage in some research situations. To make progress, scientists have contributed their enormous efforts to investigate design and semantics of quantum programming languages [1,2,3,4,5,6,7], so that quantum algorithm can be represented at relatively hight level of quantum programming languages.
In Ref. [4], D’Hondt and Panangaden introduced a notion of quantum weakest precondition and a Stonetype duality between the state transition semantics and the predicate transformer semantics for quantum programs. In their approaches, a quantum predicate is defined to be an observable, i.e., a Hermitian operator on the state space, which can be seen as a natural generalization of Kozen’s probabilistic predicate as a measurable function [12]. According to Selinger’s viewpoint [3], quantum programs may be represented by super-operators. Then, D’Hondt and Panangaden showed that quantum weakest precondition can be expressed in terms of operators of quantum programs (i.e., super-operators) and a fixed Hermitian operator [3].
Our main attention in this paper is the commutativity of quantum weakest preconditions. As the observation of Ying et al.’s [5,6] of predicate transformer semantics for classical and probabilistic programs and we should to answer some important problems that would not arise in the realm of classical and probabilistic programming. Of such problems that are known to be important, the commutativity of quantum weakest preconditions is urgent to be answered, since just as mentioned in [6], the physical simultaneous verifiability of quantum weakest preconditions depends on commutativity between them according to the Heisenberg uncertainty principle.
This paper provides three simple characterizations for commutativity of quantum weakest preconditions.
The main idea is that we should characterize the commutativity of quantum weakest preconditions in terms of the properties of quantum weakest preconditions rather than the [M, N ]. The most obvious property of quantum weakest precondition is that quantum weakest precondition is again an observable, (see Lemma 6 in the sequel), i.e. a Hermitian operator on the state space, although we often forgot this fact in practice.
Let wp(E)(M ) and wp(E(N )) be two quantum weakest preconditions. Then, this paper will show in Section 3 the following • wp(E)(M ) and wp(E)(N ) commute if and only if the product of them is Hermitian;
• wp(E)(M ) and wp(E)(N ) commute if and only if there exists an Unitary matrix U such that
where λ i and µ i are the eigenvalues of wp(E)(M ) and wp(E)(N ), respectively;
• wp(E)(M ) and wp(E)(N ) commute if and only if
where wp(E)(M ) is a quantum weakest precondition.
We would like to point out that the above results seems to be trivial (thus simple). Indeed, ( 2) and ( 3) come naturally from some facts of linear algebra [13], so long as the reader recalls that a quantum weakest precondition is again a Hermitian matrix. One may naturally expect that whether we can characterize the commutativity of quantum weakest preconditions in terms of [M, N ] (=M N -N M ), because a quantum weakest precondition is represented by a Hermitian matrix and operators of a super-operator (see, Proposition 1) (Perhaps there are some other reasons). However, the examples illustrated in this paper show that this may be very difficult (see, Proposition 7 and Proposition 8).
The remainder of the paper is organized in the following way: the next Section is devoted to review some basic definitions and useful propositions where the main results are introduced. Section 3 is devoted to the proofs of the main results where some examples are presented, and Section 4 is the concluding Section.
Let H be a Hilbert space. Recall in [6] that a density matrix ρ on H is a positive operator with where A = (a ij ) n×n .
Analogue to [6], the set of super-operators on H is denoted as CP(H).
A quantum predicate on H is defined to be a Hermitian operator M with 0 M I, where the ordering
The following intrinsic characterization of wp(E), attributed to Ying et al. [6] (also, [5]), deals with the case that E is given by a system-environment model 5 .
Proposition 2 ([6], Proposition 2.2). If E is given in terms of system-environment model, then we have
for each M ∈ P(H), where I E is the identity operator in the environment system.
and
where λ i and µ i are the eigenvalues of wp(E)(M ) and wp(E)(N ), respectively.
The following Proposition borrowed from [13] is an another characterization for commutativity of wp(E)(M ) and wp(
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
