Unitary operations are a fundamental component of quantum algorithms, but they seem to be far more useful if given with a "quantum control" as a controlled unitary operation. However, quantum operations are not limited to unitary operations. Nevertheless, it is not a priori clear if a controlled form of these general deterministic quantum operations can be well-defined. To provide a novel tool in the toolbox for quantum programming, we propose a mathematically consistent definition of a controlled form of deterministic but non-unitary quantum operations and, more generally, of quantum combs. We propose a "neutralization" comb, which transforms a set of input quantum operations to the identity operation, and study its controlled form based on our definition. We propose two new quantum algorithms for universal controllization of divisible unitary operations utilizing the most coherently controlled neutralization combs.
Unitary operations are a fundamental component of quantum algorithms, but they seem to be far more useful if given with a "quantum control" as a controlled unitary operation. However, quantum operations are not limited to unitary operations. Nevertheless, it is not a priori clear if a controlled form of these general deterministic quantum operations can be well-defined. To provide a novel tool in the toolbox for quantum programming, we propose a mathematically consistent definition of a controlled form of deterministic but non-unitary quantum operations and, more generally, of quantum combs. We propose a "neutralization" comb, which transforms a set of input quantum operations to the identity operation, and study its controlled form based on our definition. We propose two new quantum algorithms for universal controllization of divisible unitary operations utilizing the most coherently controlled neutralization combs.
Quantum supermaps are transformations that map quantum operations to quantum operations.It is known that quantum supermaps which respect a definite, predefined causal order between their input operations correspond to fixed-order quantum circuits, also called quantum combs.A systematic understanding of the physical interpretation of more general types of quantum supermaps-in particular, those incompatible with a definite causal structure-is however lacking.In this paper, we identify two types of circuits that naturally generalize the fixed-order case and that likewise correspond to distinct classes of quantum supermaps, which we fully characterize.We first introduce "quantum circuits with classical control of causal order," in which the order of operations is still well defined, but not necessarily fixed in advance: it can, in particular, be established dynamically, in a classically controlled manner, as the circuit is being used.We then consider "quantum circuits with quantum control of causal order," in which the order of operations is controlled coherently.The supermaps described by these classes of circuits are physically realizable, and the latter encompasses all known examples of physically realizable processes with indefinite causal order, including the celebrated "quantum switch."Interestingly, it also contains other examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process.Nevertheless, we show that quantum circuits with quantum control of causal order can only generate "causal" correlations, compatible with a well-defined causal order.We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how the characterizations derived in this work allow us to identify advantages for quantum information processing tasks that could be demonstrated in practice.
The quantum switch is a quantum process that creates a coherent control between different unitary operations, which is often described as a quantum process which transforms a pair of unitary operations (U1,U2) into a controlled unitary operation that coherently applies them in different orders as |0⟩⟨0|⊗U1U2+|1⟩⟨1|⊗U2U1 . This description, however, does not directly define its action on non-unitary operations. The action of the quantum switch on non-unitary operations is then chosen to be a ``natural'' extension of its action on unitary operations. In general, the action of a process on non-unitary operations is not uniquely determined by its action on unitary operations. It may be that there could be a set of inequivalent extensions of the quantum switch for non-unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case. In other words, contrary to the general case, the action of the quantum switch on non-unitary operations (as a linear and completely CP preserving supermap) is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of single-slot processes which are completely defined by their action on unitary operations.
In contrast to a classical bit, which can only take the value 0 or 1, its quantum counterpart -- a qubit -- can exist in a superposition of 0 and 1. This is a superposition of data values, naturally raising the question of whether one can superpose not only data but also programs. For example, a particular superposition of programs, known as the quantum SWITCH, has attracted much attention, and its implementations and computational advantages have been studied extensively within the physics community.
A naive way to control a program by a qubit is by means of a controlled operation. Given an operation F, this amounts to considering an operation that behaves as F when the control qubit is ∣1⟩, and as the identity operation when the control qubit is ∣0⟩. This idea works well when F is a unitary operation, but it is not well-defined for a general quantum channel. By contrast, the quantum SWITCH is free from the well-definedness issue. This contrast leads to the key insight of this paper: controlled operations and the quantum SWITCH should be regarded as different kinds of quantum control mechanisms.
Building on this insight, we develop a novel quantum programming language with quantum control and measurement that can express the quantum SWITCH over quantum channels. Using a semantic analysis based on program transformations, we identify the source of the ill-behavedness of controlled operations as the \emph{correspondence problem}: a lack of coordination between the measurements performed in the then- and else-branches of quantum conditional branching. We address this problem with a linear type system that enforces alignment of the quantum operations used in the two branches, yielding a well-behaved language capable of expressing the quantum SWITCH.
Coherent control, aka quantum control, is a central concept in quantum computing that is attracting increasing attention from both the quantum foundations and quantum software communities. Defining coherent control in the presence of recursion and measurement has long been known to be a major challenge. In particular, no-go results have been established for standard semantical domains like completely positive maps. We address this problem by introducing the first quantum programming language with recursion that allows for the coherent control of arbitrary quantum operations. We equip this language with both an operational and a denotational semantics that we prove to be adequate. To design these semantics, we show that combining coherent control, recursion, and measurement crucially requires describing the evolution of subprograms in the absence of input. To address this, the operational semantics takes into account a default evolution branch, while the denotational semantics uses the concept of coherent quantum operation, based on vacuum extensions. We strengthen the validity of our approach by developing an observational equivalence: two programs are equivalent if their probability of termination is the same in any context. The denotational semantics is shown to be fully abstract with respect to this observational equivalence.
We construct a universal quantum computer following Deutsch's original proposal of a universal quantum Turing machine (UQTM). Like Deutsch's UQTM, our machine can emulate any classical Turing machine and can execute any algorithm that can be implemented in the quantum gate array framework but under the control of a quantum program, and hence is universal. We present the architecture of the machine, which consists of a memory tape and a processor and describe the observables that comprise the registers of the processor and the instruction set, which includes a set of operations that can approximate any unitary operation to any desired accuracy and hence is quantum computationally universal. We present the unitary evolution operators that act on the machine to achieve universal computation and discuss each of them in detail and specify and discuss explicit program halting and concatenation schemes. We define and describe a set of primitive programs in order to demonstrate the universal nature of the machine. These primitive programs facilitate the implementation of more complex algorithms and we demonstrate their use by presenting a program that computes the NAND function, thereby also showing that the machine can compute any classically computable function.
An essential element of classical computation is the ``if-then'' construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two-dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.