1 Introduction
According to Moore’s law [31], within a few years, we will step into the age of quantum information and quantum computers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Quantum computers allow us to solve problems more efficiently than ever would be possible with traditional computers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The power of quantum computing is based on the fundamentals of quantum mechanics. In a quantum computer, information is represented by quantum information, and information processing is achieved by quantum gates that realize quantum operations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. These quantum operations are performed on the quantum states, which are then outputted and measured in a measurement phase. The measurement process is applied to each quantum state where the quantum information conveyed by the quantum states is converted into classical bits. Quantum computers have been demonstrated in practice [1, 2, 3, 4, 5, 6, 7, 8, 9], and the current laboratory implementations represent a promising way for quantum computers to reach store shelves soon [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19].
In the physical layer of a gatemodel quantum computer, the device contains quantum gates, quantum ports (of quantum gates), and quantum wires for the connection of the quantum circuit. The quantum gates are positioned in a quantum circuit such that several hardware constraints have to be satisfied. In contrast to traditional automated circuit design [24, 25, 26, 27, 28, 29, 30], a quantum system cannot participate in more than one quantum gate simultaneously. As a corollary, the quantum gates of a quantum circuit are applied in several rounds in the physical layer of the quantum circuit [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19].
The physical layout design and optimization of quantum circuits have different requirements with several open questions and currently represent an active area of study [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19]. Assuming that the goal is to construct a reduced quantum circuit that can simulate the original system, the reduction process has to be taken on the number of input quantum states, gate operations of the quantum circuit, and the number of output measurements. Another important question is the maximization of objective functions associated with arbitrary computational problems that are fed into the quantum circuits of the quantum computer. These parallel requirements must be satisfied simultaneously, which makes the optimization procedure difficult and are an emerging issue in present and future quantum computer developments.
In our proposed Quantum Triple Annealing Minimization (QTAM) method the goal is to determine a connection topology for the quantum circuits of quantum computer architectures that can solve arbitrary computational problems such that the quantum circuit is minimized in the physical layer, and the objective function of an arbitrary selected computational problem is maximized. The physical layer minimization covers the simultaneous minimization of the quantum circuit area (quantum circuit height and depth of the quantum gate structure), the total area of the quantum wires of the quantum circuit, the reduction of the Hamiltonian operator, and the minimization of the required number of input quantum systems and output measurements. An important aim of the physical layout minimization is that the resulting quantum circuit has to be identical to an original ‘reference’ quantum circuit (i.e., the reduced quantum circuit has to be able to simulate a nonreduced, reference quantum circuit). This simulation requirement is established by several different constraints in our automated connection topology construction procedure.
The minimization of the total quantum wire length in the physical layout is also an objective in QTAM. It serves to improve the routing quality in the topology of the quantum circuit. However, besides the minimization of the physical layout of the quantum circuit, the quantum computer also has to solve difficult computational problems very efficiently (such as the maximization of an arbitrary combinatorial optimization objective function
[16, 17, 18, 19]. To achieve this goal in our QTAM method, we also defined an objective function that provides the maximization of an arbitrary combinatorial optimization objective function. The optimization method can be further tuned by specific input constraints on the connection topology of the quantum circuit (paths in the quantum circuit, organization of quantum gates, required number of rounds of quantum gates, required number of measurement operators, Hamiltonian minimization, entanglement between quantum states, etc.) or other hardware restrictions of quantum computers, such as the wellknown nocloning theorem [22]. The various restrictions on quantum hardware, such as the number of rounds required to be integrated into the quantum gate structure, entanglement generation between the quantum states, and multipartite entanglement groups are included in our scheme. These constraints and design attributes can be handled in our scheme through the definition of arbitrary constraints on the topology of the quantum circuit via the distribution properties of the condensate wave function amplitude and phase values, or by constraints on computational paths of the quantum circuit topology.The combinatorial objective function is measured on a computational basis, and an objective function is determined from the measurement result to quantify the current state of the quantum computer. As has been demonstrated, quantum computers can be used for combinatorial optimization problems. These procedures aim to use the quantum computer to produce a quantum system that is dominated by computational basis states such that a particular objective function is maximized. In our procedure, the objective function subject of a maximization can be an arbitrary combinatorial problem.
Presently, without loss of generality, the recent experimental realizations of quantum computers are qubit architectures
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], and the current quantum hardware approaches focus on qubit systems (i.e., the dimension of the quantum system is two, ). However, while the qubit layout is straightforwardly inspirable by ongoing experiments, our method is developed for arbitrary dimensions to make it applicable for future implementations. Motivated by these assumptions, we therefore would avoid the term ‘qubit’ in our scheme to address the quantum states and instead use the generalized term, ‘quantum states’ throughout, which refers to an arbitrary dimensional quantum system. We demonstrate the results through superconducting quantum circuits [1, 2, 3, 4, 5]; however, the framework is general and flexible, allowing a realization for near term gatemodel quantum computer implementations.The novel contributions of this paper are as follows:

We define an automated method for designing quantum circuits for gatemodel quantum computers.

We conceive the QTAM algorithm, which provides an automated quantum circuit minimization on the physical layout (circuit depth and area), quantum wire length minimization, Hamiltonian minimization, input size and measurement size minimization for quantum circuits.

We define a multilayer structure for quantum circuit computations using the hardware restrictions on the connection topology of gatemodel quantum computers.
This paper is organized as follows. In Section 2, the system model is proposed. In Section 3, Section 4 and Section 5, we give the details of the method. Finally, in Section 6, we conclude the paper. Supplemental information is inlucded in the Appendix.
2 System Model
The simultaneous physicallayer minimization and the maximization of the objective function are achieved by our Quantum Triple Annealing Minimization (QTAM) algorithm. The QTAM algorithm utilizes the framework of simulated annealing (SA) [24, 25, 26, 27, 28, 29, 30], which is a stochastic pointtopoint search method.
The procedure of the QTAM algorithm with the objective functions are depicted in Fig. 1. The detailed descriptions of the methods and procedures are included in the next sections.
2.1 Computational Model
By theory, in an SAbased procedure a current solution is moved to a neighbor
, which yields an acceptance probability
[24, 25, 26, 27, 28, 29, 30](1) 
where and represent the relative performances of the current and neighbor solutions, while is a control parameter, , where is the temperature decreasing rate, is the iteration counter, is a scaling factor, while is an initial temperature.
Since SA is a probabilistic procedure it is important to minimize the acceptance probability of unfavorable solutions and avoid getting stuck in a local minima.
Without loss of generality, if is low, (1) can be rewritten in function of and as
(2) 
In the QTAM algorithm, we take into consideration that the objectives, constraints, and other functions of the method, by some fundamental theory, are characterized by different magnitude ranges [24, 25, 26, 27, 28, 29, 30]. To avoid issues from these differences in the QTAM algorithm we define three annealing temperatures, for objectives, for constraints and
for the probability distribution closeness (distance of the output distributions of the reference quantum circuit and the reduced quantum circuit).
In the QTAM algorithm, the acceptance probability of a new solution at a current solution is as
(3) 
where , and are the average values of objective, constraint and distribution closeness domination, see Algorithm 1.
To aim of the QTAM algorithm is to minimize the cost function
(4) 
where
is the vector of design variables, while
is the vector of weights, while is the number of primarily objectives. Other secondary objectives (aspect ratio of the quantum circuit, overlaps, total net length, etc.) are minimized simultaneously via the singleobjective function in (4) as(5) 
2.2 Objective Functions of QTAM
We defined objective functions for the QTAM algorithm. Objective functions and are defined for minimization of quantum circuit in the physical layer. The aim of objective function is the minimization of the quantum circuit area of the quantum gate structure,
(6) 
where is the optimal circuit height of , while is the optimal depth of .
Focusing on superconducting quantum circuits [1, 2, 3, 4, 5], the aim of is the physical layout minimization of the total quantum wire area of , as
(7) 
where is the number of nets of the circuit, is the number of quantum ports of the quantum circuit considered as sources of a condensate wave function amplitude [1, 2, 3, 4, 5], and the number of quantum ports considered as sinks of a condensate wave function amplitude, is the length of the quantum wire , is the effective width of the quantum wire , while is the (root mean square) condensate wave function amplitude [1, 2, 3, 4, 5] associated to the quantum wire .
Objective function is defined for the maximization of the expected value of an objective function as
(8) 
where is an objective function, is a collection of parameters
(9) 
such that with unitary operations, state is evaluated as
(10) 
where is an th unitary that depends on a set of parameters , while is an initial state. Thus the goal of is to determine the parameters of (see (9)) such that is maximized.
Objective functions and are defined for the minimization of the number of input quantum states and the number of required measurements. The aim of objective function is the minimization of the number of quantum systems on the input of the circuit,
(11) 
The aim of objective function is the minimization of the total number of measurements in the measurement block,
(12) 
where , where is the number of measurement rounds, is the number of measurement gates in the measurement block.
2.3 Constraint Violations
The optimization at several different objective functions results in different Pareto fronts [24, 25, 26, 27] of placements of quantum gates in the physical layout. These Pareto fronts allow us to find feasible tradeoffs between the optimization objectives of the QTAM method. The optimization process includes diverse objective functions, constraints, and optimization criteria to improve the performance of the quantum circuit and to take into consideration the hardware restrictions of quantum computers. In the proposed QTAM algorithm the constraints are endorsed by the modification of the Pareto dominance [24, 25, 26, 27] values by the different sums of constraint violation values. We defined three different constraint violation values.
2.3.1 Distribution Closeness Dominance
In the QTAM algorithm, the Pareto dominance is first modified with the sum of distribution closeness violation values, denoted by . The aim of this iteration is to support the closeness of output distributions of the reduced quantum circuit to the output distribution of the reference quantum circuit .
Let the output distribution after the measurement phase of the reference (original) quantum circuit to be simulated by , and let be the output distribution of the actual, reduced quantum circuit . The distance between the quantum circuit output distributions and (distribution closeness) is straightforwardly yielded by the relative entropy function, as
(13) 
For two solutions and , the distribution closeness dominance function is defined as
(14) 
where is evaluated for a given solution as
(15) 
where is an th distribution closeness violation value, is the number of distribution closeness violation values for a solution .
In terms of distribution closeness dominance, dominates if the following relation holds:
(16) 
thus (16) states that dominates if both and are unfeasible, and is closer to feasibility than , or is feasible and is unfeasible.
By similar assumptions, dominates if
(17) 
2.3.2 Constraint Dominance
The second modification of the Pareto dominance is by the sum of constraint violation values,
(18) 
where is the sum of all constraint violation values, evaluated for a given solution as
(19) 
where is an th constraint violation value, is the number of constraint violation values for a solution .
Similar to (16) and (17), in terms of constraint dominance, dominates if the following relation holds:
(20) 
thus (16) states that dominates if both and are unfeasible, and is closer to feasibility than , or is feasible and is unfeasible.
By similar assumptions, dominates with respect to if
(21) 
2.3.3 Objective Dominance
Let and refer to two solutions, then, by theory, the objective dominance function is defined as
(22) 
where is the number of objectives (in our setting ), is the range of objective , while dominates if for , and for at least one the relation holds.
2.4 Quantum Circuit Design Constraints
Since it is a hard problem to find a single placement solution in the physical layer that satisfies all design constraints, constrained multiobjective optimization methods were proposed to handle the situation. The term proximity group (a group of cells with a symmetry axis and different cell types with varying symmetry requirements) is used in analog floorplan automation [24, 25, 26, 27, 28, 29, 30] to reduce the problem complexity. Taking the Pareto fronts of placements of the proximity groups, a tradeoff can be constructed between the optimization objectives. These Pareto fronts also can be combined in a hierarchical way for each proximity group. Further constraints on the condensate wave function amplitude and phase are also taken to improve the quantum circuit’s quality and performance.
3 The QTAM Algorithm
Theorem 1
The QTAM algorithm utilizes annealing temperatures , and to evaluate the acceptance probabilities, where is the annealing temperature for the objectives, is the annealing temperature for the constraints and is the annealing temperature for the distribution closeness.
Proof. The detailed description of the QTAM procedure is given in Algorithm 1.
(23) 
(24) 
(25) 
(26) 
(27) 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
(35) 
The related steps are detailed in Subprocedures 14.
3.1 Computational Complexity of QTAM
Following the complexity analysis of [24, 25, 26, 27], the computational complexity of QTAM is evaluated as
(36) 
where is the number of dominance measures, is the number of total iterations, is the population size, while is the number of objectives. Therefore, at the objective functions , the resulting complexity is
(37) 
4 Wiring Optimization and Objective Function Maximization
4.1 Multilayer Quantum Circuit Grid
An th quantum gate of is denoted by , a th port of the quantum gate is referred to as . Due to the hardware restrictions of gatemodel quantum computer implementations [16, 17, 18, 19], the quantum gates are applied in several rounds. Thus, a multilayer, dimensional (for simplicity we assume ), sized finite squarelattice grid can be constructed for , where is the number of layers, , . A quantum gate in the th layer is referred to as , while a th port of is referred to as .
4.2 Method
Theorem 2
There exists a method for the parallel optimization of quantum wiring in physicallayout of the quantum circuit and for the maximization of an objective function .
Proof. The aim of this procedure (Method 1) is to provide a simultaneous physicallayer optimization and Hamiltonian minimization via the minimization of the wiring lengths in the multilayer structure of and the maximization of the objective function (see also Section A.1). Formally, the aim of Method 1 is the simultaneous realization of the objective functions and .
Using the multilayer grid of the quantum circuit determined via and , the aim of maximization of the objective function , where in an length input string, where each is associated to an edge of connecting two quantum ports. The objective function associated to an arbitrary computational problem is defined as
(38) 
where is the objective function for an edge of that connects quantum ports and .
The maximization of objective function (38) yields a system state for the quantum computer [16, 17, 18, 19] as
(39) 
where
(40) 
where is a unitary
(41) 
with , , where is the Pauli operation, is a control parameter [16, 17, 18, 19], and
(42) 
where is the dimension of the quantum system, while
(43) 
The objective function (38) without loss of generality can be rewritten as
(44) 
where each act on a subset of bits, such that . Therefore, there exists a selection of parameters of in (9) such that (44) picks up a maximized value , which yields system state as
(45) 
Therefore, the resulting Hamiltonian associated to the system state (45) is minimized via (see (57)) as
(46) 
since the physicallayer optimization minimizes the physical distance between the quantum ports, therefore the energy of the Hamiltonian associated to is reduced to a minima.
The steps of the method are given in Method 1. The method minimizes the number of quantum wires in the physicallayout of , and also achieves the desired system state of (39).
The steps of Method 1 are illustrated in Fig. 2, using the multilayer topology of the quantum gate structure, refers to the th layer of .
5 Quantum Circuit Minimization and Routing
5.1 Quantum Circuit Area Minimization
For objective function , the area minimization of the quantum circuit requires the following constraints. Let be the vertical symmetry axis of a proximity group [24, 25, 26] on , and let refer to the coordinate of . Then, by some symmetry considerations for ,
(47) 
where is the bottomleft coordinate of a cell , is the width of , and
(48) 
where is the bottomleft coordinate of a cell , is the height of .
Let be a symmetry pair [24, 25, 26] that refers to two matched cells placed symmetrically in relation to , with bottomleft coordinates . Then, can be rewritten as
(49) 
with the relation .
Let be a cell which is placed centered [24, 25, 26] with respect to . Then, can be evaluated as
(50) 
along with . Note that it is also possible that for some cells in there is no symmetry requirements, these cells are denoted by .
As can be concluded, using objective function for the physicallayer minimization of , a dimensional constraint vector can be formulated with the symmetry considerations as follows:
(51) 
where is the number of symmetry pairs, is the number of type cells, while is the number of type cells, while is the rotation angle of an th cell , respectively.
5.2 Quantum Wire Area Minimization
Objective function provides a minimization of the total quantum wire length of the circuit. To achieve it we define a procedure that yields the minimized total quantum wire area, , of as given by (7). Let be the effective width of the quantum wire in the circuit, defined as
(52) 
where is the (root mean square) condensate wave function amplitude, is the maximum allowed current density at a given reference temperature , while is the nominal layer height. Since drops in the condensate wave function phase are also could present in the circuit environment, the effective width of the quantum wire can be rewritten as
(53) 
where is a maximally allowed value for the phase drops, is the effective length of the quantum wire, while is a conductor sheet resistance [1, 2, 3, 4, 5].
In a multilayer topological representation of , the distance between the quantum ports is as
(54) 
where is a cost function between the layers of the multilayer structure of .
During the evaluation, let be the total quantum wire area of a particular net of the circuit,
(55) 
where quantum ports are considered as sources of condensate wave function amplitudes, while of are sinks, thus (7) can be rewritten as
(56) 
Since is proportional to , (56) can be simplified as
(57) 
where
(58) 
where is given in (54).
In all quantum ports of a particular net of , the source quantum ports are denoted by positive sign [24, 25, 26] in the condensate wave function amplitude, assigned to quantum wire between quantum ports and , while the sink ports are depicted by negative sign in the condensate wave function amplitude, with respect to a quantum wire between quantum ports and .
Thus the aim of in (55) is to determine a set of porttoport connections in the quantum circuit, such that the number of long connections is reduced in a particular net of as much as possible. The result in (56) is therefore extends these requirements for all nets of .
5.2.1 Wave Function Amplitudes
With respect to a particular quantum wire between quantum ports and of , let refer to the condensate wave function amplitude in direction , and let refer to the condensate wave function amplitude in direction in the quantum circuit. Then, the let be defined for the condensate wave function amplitudes of quantum wire as
(59) 
with a residual condensate wave function amplitude
(60) 
where is an actual amplitude in the forward direction . Thus, the maximum amount of condensate wave function amplitude injectable to of quantum wire in the forward direction at the presence of is (see (60)). The following relations holds for a backward direction, , for the decrement of a current wave function amplitude as
(61) 
with residual quantum wire length
(62) 
where is given in (52).
By some fundamental assumptions, the residual network of is therefore a network of the quantum circuit with forward edges for the increment of the wave function amplitude , and backward edges for the decrement of . To avoid the problem of negative wire lengths the BellmanFord algorithm [24, 25, 26] can be utilized in an iterative manner in the residual directed graph of the topology.
To find a path between all pairs of quantum gates in the directed graph of the quantum circuit, the directed graph has to be strongly connected. The strongconnectivity of the nets with the parallel minimization of the connections of the topology can be achieved by a minimum spanning tree method such as Kruskal’s algorithm [24, 25, 26].
Lemma 1
The objective function is feasible in a multilayer quantum circuit structure.
Proof. The procedure defined for the realization of objective function on a quantum circuit is summarized in Method 2. The proof assumes a superconducting architecture.
The subprocedures of Method 2 are detailed in Submethods 2.1, 2.2 and 2.3.
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