In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50
In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50
Quantum circuits If the state of a quantum system is a kind of information, then the dynamics of that system is a kind of information processing. This is the basic idea behind quantum computing , which seeks to exploit the physics of quantum systems to do useful information processing. In this chapter we will acquaint ourselves with a few of the ideas of quantum computing, using the idealized quantum circuit model. Then we will turn our attention to an actual quantum process, nuclear magnetic resonance , that can be understood as a realization of quantum information processing. In a quantum circuit, we have a set of n qubit systems whose dynamical evolution is completely under our control. We represent the evolution as a sequence of unitary operators, each of which acts on one or more of the qubits. Graphically, we represent the qubits as a set of horizontal lines, and the various stages of their evolution as a series of boxes or other symbols showing the structure of the sequence of operations. In such a diagram, time runs from left to right (see Fig. 18.1.). The n qubits form a kind of “computer,” and its overall evolution amounts to a “computation.” The key idea is that very complicated unitary operations on the n qubits can be built up step-by-step from many simple operations on one, two or a few qubits.
We are now ready to see how computing with qubits can be done. In this book, we will mainly use the circuit model for computation which was first introduced by Deutsch [25]. We will represent by quantum “wires,” the qubits upon which manipulations. The length of the wire is to be interpreted as the time axis. Manipulations on qubits can be done using basic unitary operators that are the equivalents of logic “gates.” An algorithm, or a complete set of steps for achieving a processing task, is a combination of wires and gates representing a quantum circuit. This circuit must be thought of as a time sequence of events with every wire a way of representing qubit states, and with gates representing processing of those states.
This paper explores a novel approach to quantum computation centered around self-referential quantum circuits. The core concept involves designing circuits that recursively incorporate themselves into their operational structure, effectively creating computational modules capable of self-replication. This is achieved through the strategic implementation of feedback loops and entanglement within the circuit architecture. The potential benefits of this paradigm include a reduction in circuit complexity and the possibility of constructing universal quantum computers from fundamentally simpler components. The theoretical framework presented herein outlines the key principles and mathematical underpinnings of this self-referential design, proposing a pathway toward more efficient and scalable quantum computation. The presented model uses the following notation: |ψ⟩ represents a quantum state, U(θ) represents a unitary operator with rotation angle θ, C represents a quantum circuit, and E represents the expectation value. Specifically, the expectation value of an observable A is denoted as E[A] = <ψ|A|ψ>, where |ψ> is the quantum state. The Hamiltonian of a quantum system is typically represented as H, and the time evolution of the state is governed by the Schrödinger equation: iℏ∂|ψ(t)⟩/∂t = H|ψ(t)⟩. Entanglement is represented by the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. Quantum gates, such as the Hadamard gate (H) and the CNOT gate (CN), are represented as unitary transformations. The circuit representation involves sequential application of gates, and the overall unitary transformation is the product of these gates. The circuit depth, denoted as d, represents the number of consecutive gate operations applied to a quantum state. The circuit complexity, denoted as C, is a measure of the number of gates required to implement a specific quantum algorithm. The circuit fidelity, denoted as F, quantifies the accuracy of a quantum circuit's output, ranging from 0 to 1. We define the quantum state preparation as |ψ⟩ = ∑i ci|i⟩, where ci are complex coefficients and |i⟩ are the computational basis states. The unitary transformation is defined as U|ψ⟩ = ∑i (Uici)|i⟩, where Ui are the matrix elements of the unitary operator U. The propagation of quantum states through a circuit is represented by the Schrödinger equation, and the final state is obtained by applying the unitary transformation to the initial state. The concept of quantum coherence, denoted as χ, represents the ability of a quantum system to maintain its superposition state. The fidelity of a quantum state is defined as the probability that a measurement will yield the expected outcome.
The general computational process A suitably programmed quantum computer should act on a number x to produce another number f(x) for some specified function f . Appropriately interpreted, with an accuracy that increases with increasing k , we can treat such numbers as non-negative integers less than 2 k . Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits. If we specify the numbers x as n -bit integers and the numbers f(x) as m -bit integers, then we shall need at least n + m Qbits: a set of n -Qbits, called the input register , to represent x , and another set of m -Qbits, called the output register , to represent f(x) . Qbits being a scarce commodity, you might wonder why we need separate registers for input and output. One important reason is that if f(x) assigns the same value to different values of x , as many interesting functions do, then the computation cannot be inverted if its only effect is to transform the contents of a single register from x to f(x) . Having separate registers for input and output is standard practice in the classical theory of reversible computation. Since quantum computers must operate reversibly to perform their magic (except for measurement gates), they are generally designed to operate with both input and output registers. We shall find that this dual-register architecture can also be usefully exploited by a quantum computer in some strikingly nonclassical ways.
Except for the NOT gate, all the other familiar classical gates are irreversible in the sense that we cannot uniquely reconstruct the input states from the output states. For example, OR, AND, NAND, NOR, etc. are irreversible. They all map a two-bit input state into a single bit output state. Thus one bit is erased during operation of each of these gates, and according to Landauer’s principle that requires dissipation of a minimum amount of energy. Now a simple question arises in our curious mind: Is it possible to avoid this loss of energy? The answer is yes. If we don’t erase any bit then we can circumvent this energy loss. Thus we need to map n bit input states to n bit output states. In addition, if a one-toone correspondence exists between the input states and the output states then only we will be able to uniquely reconstruct the input states from the output states. In such a case the gate is called reversible. For example, if we have f(00) = 00, f(01) = 10, f(10) = 01, f(11) = 11 then f represents a reversible gate. Quantum evolution operators are unitary so for every operator U we have an inverse operator U−1 = U†. Therefore, quantum evolution operators are the natural choice for the construction of energy efficient reversible gates. However, it is not the only choice. We can have classical reversible gates, too. Usually by reversible gates we refer to classical reversible gates, and quantum gates are specifically referred to as quantum gates. Reversible gates and quantum gates are similar but there exists a fundamental difference that reversible gates cannot accept superposition states (e.g. α|0〉 + β|1〉) as input states, whereas quantum gates can. Thus all quantum gates are essentially reversible but the converse is not true.
In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, i.e. unitary operators acting on pure states (qubits or quregisters) of a convenient (n-fold tensor product) Hilbert space ⊗nC2 (where C2 is the bidimensional complex Hilbert space). Consequently, quantum gates represent reversible time evolution of the system. According to [1] however, such a representation is unduly restrictive insomuch it does not encompass open systems where coupling with environment and measurement processes may occur. In this case, time evolution is no longer reversible. In [1] the authors formulate a more general model of quantum computational processes, where pure states and unitary operators are replaced by density operators and quantum operations, respectively. Let H be a Hilbert space, L(H) the vector space of all linear operators of H and D(H) the set of density operators in H. A quantum operation [2] is a linear and completely positive operator O: L(H1) → L(H2). By [2], for every quantum operation O: L(H1) → L(H2), there exists a set of operators {Ai} such that iA iAi = I and ∀ρ ∈ D(H1): E(ρ) = iAiρA