On the Design of Molecular Excitonic Circuits for Quantum Computing: The Universal Quantum Gates

This manuscript presents a theoretical strategy for encoding elementary quantum computing operations into the design of molecular excitonic circuits. Specifically, we show how the action of a unitary transformation of coupled two-level systems can be equivalently represented by the evolution of an exciton in a coupled network of dye molecules. We apply this strategy to identify the geometric parameters for circuits that perform universal quantum logic gate operations. We quantify the design space for these circuits and how their performance is affected by environmental noise. Abstract This manuscript presents a strategy for controlling the transformation of excitonic states through the design of circuits made up of coupled organic dye molecules. Speciﬁcally, we show how unitary transformation matrices can be mapped to the Hamiltonians of physical systems of dye molecules with speciﬁed geometric and chemical properties. The evolution of these systems over speciﬁc times encode the action of the unitary transformation. We identify the bounds on complexity of the transformations that can be represented by these circuits. We formalize this strategy and apply it to identify the excitonic circuits of the four universal quantum logic gates: NOT, Hadamard, π/ 8 and CNOT. We discuss the properties of these circuits and how their performance is expected to be inﬂuenced by the presence of environmental noise. We quantify the bounds on the spectroscopic properties of organic dye circuits under which single-qubit unitary transformations are possible.


Introduction
The elementary component of a quantum computer -a qubit -is a two-state quantum system. A qubit can be constructed from many different physical systems, including a pair of coupled organic dye molecules sharing a single electronic excitation (i.e., an exciton).
Using this kind of qubit it is therefore possible, at least in principle, to develop quantum computing platforms that operate via the excited state dynamics of specifically designed excitonic circuits comprised of multiple dye molecules in precise geometric arrangements. In this manuscript, we introduce a general strategy for designing excitonic circuits for quantum computation. We apply this strategy to identify fundamental bounds on the computational complexity that these circuits can support and identify the physical requirements for performing universal quantum logic gate operations on one-and two-qubit systems. This study therefore sets the groundwork for enabling the development of programmable dye-based quantum computing platforms. Supermolecular support structures, such as proteins, 2,3 metal-organic frameworks, 4 and DNA nanostructures, 5,6 can be used to situate dye molecules with coupling networks that are designed to control certain aspects of exciton dynamics. The resulting dynamical control can be used to implement the state transformations required for quantum computation.
Quantum computing offers several key advantages over traditional classical computing and is poised to make a transformative impact on certain areas of the information sciences, such as cryptography and molecular simulation. 7,8 However, despite enormous potential for broad technological impact, quantum computing presents unique implementation challenges that have thus far limited it to only a few physical systems. 9 This includes optical cavi-ties, 10,11 trapped ions, 12-14 molecular spins, 15,16 superconductors, 17,18 quantum dots 19,20 and solid state color centers. 21,22 From the standpoint of quantum computing, each of these qubit systems have their own strengths and limitations. Practical application of any specific system will require exploiting its strengths while mitigating its limitations. Characterizing the strengths and limitations of new potential qubit systems, such as those made from excitonic circuits, is an important step in the development of quantum information technologies.
A limitation that affects nearly all qubit systems is the requirement for low operating temperatures. This requirement is intrinsic to the physics of some systems, such as superconducting and trapped atom qubits. Low temperature is also used to reduce the effects of environmental noise, which can destroy the delicate phase information required for quantum computation. Unfortunately, achieving and maintaining the low temperatures that are required for these systems is both expensive and impractical. Qubit systems with the ability to maintain and share their phase information in noisy thermal environments could significantly improve the scalability of quantum computing technologies. 23,24 The dye molecules that comprise excitonic circuits are highly sensitive to environmental noise and exhibit coherence times that are generally much shorter than existing qubit systems. On the other hand, the dye molecules can be strongly coupled so as to enable rapid transfer and evolution of phase information. Viable quantum computing in excitonic circuits will require balancing short coherence times with the ability to produce strong intermolecular electronic couplings. Theoretical models, such as we present here, play an important role in developing an understanding of this balance and its potential implications.
The focus of this paper is to explore the opportunities and limitations in engineering quantum dynamics in excitonic molecular systems. We show that these dynamics can be programmed and exploited to realize a broad range of quantum operations, including, in particular, a universal set of quantum logic gates. 25,26 Promisingly, coherent dynamics in excitonic molecular systems are seen to survive moderate levels of environmental noise, suggesting they may be candidates for a new class of quantum materials with information processing applications.
We present a general strategy for programming the dynamics of excitons in the design of excitonic circuits. Importantly, this programming enables the implementation of any unitary transformation, such as those that form the basis for quantum information processing. To accomplish this implementation we map qubit states onto the excitation states of coupled dye molecules. For example, the basis states of a two qubit system, i.e., {|00 , |01 , |10 , |11 }, can be mapped onto a basis of localized single molecule excitations in an excitonic circuit made of four molecules. The excitonic representation of a qubit system naturally supports quantum properties such as superposition, i.e., through excitonic delocalization, and can encode the coherence and entanglement of the two qubit system.
In the following section we present the details of this strategy and its application to the set of universal quantum logic gates. Then, in Section 3, we simulate the performance of these gates under varying environmental conditions.

Frenkel Exciton Model
It is convenient to describe the excited state of a N-molecule system in a reduced basis of single molecule excitations. If one then assumes that each molecule can only access one ground and one excited state, then the basis becomes that of the Frenkel exciton model. 27,28 Specifically, the generic Hamiltonian of the Frenkel model is given by, where |i is the basis state where molecule i is in the excited state (with all other molecules in the ground state), i is the energy of this basis state, and V ij is the electronic coupling between the states |i and |j .
This simple model has been widely used in the study of excited multi-chromophoric systems due to its computational efficiency. 29,30 Although the Frenkel model omits influence of higher order excitations, many-body effects, nuclear relaxation, and the specific details of molecular electronic structure, it has been found to be remarkably accurate for reproducing the results of experimental and higher level theory when appropriately parametrized. We thus employ the Frenkel exciton model in this study, acknowledging that the systems we describe below can be described using a higher level model in subsequent studies, if missing.
We assume that the configurations of dye molecules in the system are defined by their center of mass positions and orientations. We assume that the coupling between dye molecules is given by the point dipole approximation, where µ i is the transition dipole moment of excitation |i , r ij = r i − r j is the displacement vector between the dyes,r ij = r ij /r ij is the corresponding displacement unit vector and ε 0 is the vacuum permitivity. Again, this coupling can be quantified with a less approximate expression if higher accuracy is needed.
Despite its simplicity, the Frenkel exciton model encodes all of the information of the system dynamics required for the scope of this paper. As we will see in the following sec-

Engineering Exciton Dynamics
Molecular excitonic systems with tunable geometry present a unique opportunity to design systems that realize specific quantum transformations. This approach is illustrated in Figure   1 for schematic quantum circuits. In particular, closed system quantum dynamics generate a family of unitary transformations, {Û (t) ≡ e − i Ĥ t }, from the system Hamiltonian,Ĥ.
A system Hamiltonian,Ĥ, can be represented by a target unitary transformation,Û target , where τ gives the transformation time at whichÛ target is realized. The family of Hamiltonians, {Ĥ τ }, are scalar multiples that differ only in the transformation time.
Clearly Eq. (4) is satisfied for all N < ∼ 12 leaving at least three excess degrees of freedom.
These excess degrees of freedom indicate the existence of a manifold of possible dye configurations that implement a given transformation. As we will discuss later, this extra flexibility can be used to optimize the performance of the system in the presence of environmental noise and to construct systems that enable straightforward experimental initialization and measurement.
This analysis bounds the complexity of transformations that can, in principle, be realized in excitionic systems. However, it assumes a freedom in selecting the positioning, energies and transition dipole moments of the dyes that does not exist in practical applications.
Realistically, any given scaffolding approach may have restrictions on dye placement and may only be compatible with a restricted subset of dyes. These practical constraints make the added flexibility of the excess degrees of freedom essential as they can be used to design around the limitations of a given experimental approach. In the remainder of this paper we will show that, at least for simple transformations, the required constraints leave a great deal of flexibility of implementation.

Example: Universal Quantum Gates
We now illustrate the use of this approach by considering the implementation of a universal set of quantum gates in a dye system. These simple transformations are widely studied as the building blocks of all quantum algorithms. We will show that the one-qubit NOT, Hadamard, and π/8 transformation, and the entangling two-qubit CNOT transformation (shown in Table 1) can be implemented in excitonic dye systems. Moreover, we find that excitonic circuits have far more flexibility than required to realize these transformations, allowing us to design around practical limitations (e.g. limited dye libraries), optimize performance in the presence of noisy environments and even design systems that are easier to prepare and measure with a given experimental set up.
Each state |i can be identified in the site basis by the dye molecule where the excitation is localized. Each of these dyes is then associated with a state of the qubit register, mapping the register state to the exciton location. We will then denote each dye by the qubit state from which it is mapped. For the one qubit gate this maps the qubit states |0 and |1 to the two dyes |A and |B , respectively, while for two qubit gates the qubit states |00 , |01 , |10 and |11 mapped to the four dyes |A , |B , |C and |D , respectively The simplest of these transformations is the π/8 phase gate. This gate increases the relative phase between states |0 and |1 giving the operationÛ π/8 and corresponding Hamiltonian in Table 1. This Hamiltonian leads to two intuitive constraints on the dye assembly -one on the coupling and the other one on the relative excitation energy of the dyes. First, since this gate does not induce transitions between qubit states, the dyes must be uncoupled, i.e. V 01 = 0. Second, to allow the two states to acquire relative phase, the dyes must be non-degenerate with ∆ 01 = 0 − 1 = 0. These constraints are satisfied by any uncoupled dye heterodimer allowing for any pair of nondegerate dyes and a broad range of possible geometric configurations. Moreover, the transformation time is given by τ = π/(4∆ 01 ).
We now consider the single-qubit NOT gate, represented by the unitary operationÛ NOT , in Table 1. This transformation swaps qubit states |0 and |1 without modifying their relative phase. To ensure excitation transfer the two dyes must have non-zero coupling (i.e. V 01 = 0). In addition, they must also be degenerate to ensure that the excitation fully transfers between the states. Thus, any coupled pair of degenerate dyes, e.g., a homodimer, reproduces a quantum NOT gate. Physically, this exploits the oscillatory energy transfer in a homodimer to coherently swap qubit states, where the NOT operation is realized at half the Rabi frequency, τ = π/(2V 01 ), when population inversion is maximized.
In a similar way, we can identify a molecular system that represents the action of a Hadamard gate on an input qubit state. This gate is represented by the operator,Û Had (Table 1). The Hadamard gate transforms an initial state into a superposition of the qubit states |0 and |1 . The system described byĤ Had corresponds to a heterodimer coupled by V 01 = π/(2 √ 2)τ , where the relative transition energies of the dyes are given by ∆ 01 = π/( √ 2τ ). We see thatĤ Had impose an additional constrain on the system as the ratio of the energy difference, ∆ 01 , and the coupling between the dyes, V 01 , must be equal to a constant factor: ∆ 01 /V 01 = 2. Consequently, the coupling between the dyes in the heterodimer will be completely specified after we choose a value for ∆ 01 , and only a reduced set of dye spatial distributions will evolve with a HamiltonianĤ Had for such a system.
The CNOT gate is an operator that acts over two qubits: one control qubit and one target qubit. If the control qubit is set to zero, then the operator does not act over the target qubit, but if the control qubit is set to 1, then the CNOT operator acts over the target as a NOT gate. The CNOT gate is represented by the 4 × 4 evolution operator,Û CNOT (Table 1). The control operation is represented in the upper left quarter ofÛ CNOT , a 2 × 2 identity matrix, while the NOT operation is represented in the lower right quarter ofÛ CNOT .
The Hamiltonian that corresponds to this operator is given byĤ CNOT . Correspondingly, a CNOT gate can be realized by a coupled homodimer representing the NOT operation, and two identical uncoupled dyes corresponding to the identity operation.
Since the CNOT operation does not change the relative phase between any of the states, the excitation energy of the control and target dyes must be selected such that ∆ 01 = π/(2τ ) = V 01 .

Selective Excitation and Measurement Schemes
In pulses. Each of these pairs can then be oriented orthogonally to each other with |00 and |10 orthogonal to |01 and |11 . This arrangement then allows for a polarization addressing of the target qubit state, analogous to the two-dimensional approach.
These addressing strategies are of course not unique. However, they demonstrate how the excess degrees of freedom can be exploited to satisfy additional constraints imposed by experimental limitations (e.g. orthogonality for polarization addressing). Moreover, these constraints can be softened to satisfy other technical constraints. For example, polarization addressed dyes can be placed in non-orthogonal configurations. While this reduces the specificity of the addressing procedure, it can increase the coupling between the dyes, mitigating the effect of environmental noise. In the following section, we explore how dye configurations can be tuned to optimize this type of trade off.

Open System Dynamics
Thus far, we have restricted our attention to closed system dynamics where the state of the quantum system can be represented as a linear combination of the form ψ = a 0 |0 + a 1 |1 .
In this setting, the system is isolated from the surroundings and retains all of its phase information as it evolves in time. However, when a system evolves in contact with a bath, environmental noise and the formation of uncontrolled system-bath entanglement leads to the gradual loss of phase information of the system. In the presence of this incomplete phase information, the state of the quantum system can no longer be represented by a wavefunction. Instead,the system state must be represented by a density matrix ρ. 33 In this matrix representation, the diagonal components, ρ ii , give the population of state |i , playing an equivalent role to the probability amplitudes |a i | 2 . The complex-valued off diagonal components, ρ ij , are known as coherences and describe the phase information between states |i and |j .
In the following sections, we will consider the evolution of the one qubit gates in the presence of a noisy environment. This will allow us to examine the limitations and requirements of excitonic quantum information processing and illustrate the optimization of these quantum circuits. By taking advantage of the normalization condition, |α| 2 + |β| 2 = 1, the density matrix of a two-level qubit can be conveniently represented by the density matrix where σ x , σ y and σ z are real valued components of a 3D vector σ known as the Bloch vector.
The properties of the density matrix constrain this vector to the sphere | σ| ≤ 1, which is referred to as the Bloch sphere.
In this compact representation, the evolution of a closed system reduces to solving the set of differential equations for the Bloch vector σ, known as the Liouville-von Neumann In an open system, interaction with a noisy environment substantially modifies the system dynamics. Generally, these dynamics can be quite complicated, potentially showing substantial non-Markovian character that depends intricately on the structure and statistics of the environment. In this study, we aim to consider a simple model for the influence of the bath that relies minimally on the details of the local environment. As such, we will restrict our attention to simple phenomenological Markovian master equations of the Lindblad type. 33 In this model two major effects are included for a system of dye molecules coupled to a phonon bath. The dephasing, with rate γ, describes random fluctuations in the energy levels of the dye molecules due to environmental noise. This leads to the loss of coherent phase information, manifest in a decay in the off-diagonal components of ρ, or coherences.

equations. For a general Hamiltonian of the form H
In addition, dissipation, with rate Γ describes the loss of excitation energy to the phonon bath as the system relaxes to the lower energy eigenstate. Including these dephasing and dissipation effects in Eq. (6), we arrive at an expression for the Bloch equations in an open system:σ The quantity T 2 = 2/(Γ + 2γ) is often referred to as the total dephasing time while T 1 = Γ −1 is called the dissipation or relaxation time. As in the case of the Frenkel exciton model described in Section 2.1, this simple model is used with the acknowledgement that more sophisticated approaches to these dynamics may be required to treat specific systems in future work.
In the following section, we will use Eq. (7) to model the dynamics of the NOT and Hadamard gates in order to illustrate the effects of environmental noise on the desired unitary transformation. For simplicity, we will restrict our attention to dyes with linear (as opposed to circular) transition dipole moments. This leads to a real valued coupling between the The spatial dependence of the inter-dye coupling can then be expressed, according to Eq. (2), in terms of a set of three parameters: the inter-chromophore distance r ij , the twist angle θ ij and the dye-pair angle relative to the distance vectorr ij , ϕ.

Gate Performance in an Environment
The dynamics of the NOT and Hadamard quantum gates under the effect of phonon bath can be derived by solving the system of equations in Eq. (7), using the appropriate Hamiltonian in Table 1. The relative populations of the states of the two-level qubit system in the site basis, as well as the coherences at a a given time, t, can be extracted from the density matrix in Eq. (5). As an example, we examine the population dynamics of the NOT gate. Figure   2A shows the population dynamics of the state |1 under the effect of different dephasing rates. In the isolated case, the required state inversion of the input qubit (taking the qubit from |0 to |1 and from |1 to |0 ), is implemented by Rabi oscillations with a period 2τ .
This first maximum corresponds to the time it takes the system to perform a single gate operation, before returning again to its initial state, at t = 2τ . The oscillatory nature of the dynamics indicates that for a given configuration the NOT gate transformation is in fact realized at many times τ n = nτ , where n is any odd integer. The oscillations in Figure 2A are seen to decrease rapidly with increasing dephasing rate γ, since the coherence is almost completely lost when the dephasing time, 1/γ, is 1/3 of τ .
The performance of a quantum gate can be quantified according to the schematic in Fig. 2B. In a closed system, a perfect NOT gate would interconvert 100% of the initial qubit between |0 and |1 . However, dephasing (here shown for γ ∼ τ ) reduces the amplitude of the Rabi oscillations, decreasing the amount of the excitation transferred to the desired state. At longer times, dephasing fully damps the oscillations leaving an equal (incoherent) mixture of |0 and |1 states. We therefore want to measure the state of the system at the minimum time it requires to perform the desired operation, τ , and the efficiency must be determined at t = τ (dotted red line in Figure 2B).
Following this idea, we define a fidelity measure for a two state excitonic circuit that quantifies the probability of measuring the correct outcome after the transformation is ap-plied. This quantity includes the deviation of the state of the open system from that of the closed quantum system at time t = τ and the ability to perform the polarization addressing scheme proposed in Section 2.4. Describing the state of the system at a given time by the density matrix, ρ, in Eq. (5), the fidelity of the circuit can be defined by a system well isolated from environmental noise, the result is that of a pure state density matrix Tr{ρ 2 closed } = 1. 33 The last term, sin 2 θ ij , describes the ability of the polarization addressing scheme to distinguish between a pair of dyes when the system is measured. This gives a vanishing fidelity for parallel and anti-parallel dyes (i.e. θ ij = 0, π) since the two dyes cannot be distinguished by a polarized pulse in these configurations. rameter regime for reliable implementation of two dimensional unitary transformations. To illustrate the effect of dissipation and dephasing on the fidelity of a NOT gate circuit, we use as an example a pair of nearly-orthogonal Cy3 dyes scaffolded in adjacent nucleotide bases: 6 r ij /µ ≈ 0.27Å/D, θ ij = 4π/9 and ϕ = π/2. The effect is shown in Fig. 3A.
As expected from the dynamics shown in Fig. 2 When (Γ + γ)τ > 1, the system has lost its coherence before the gate operation is completed for the first time. Consequently, an excitonic NOT and Hadamard gate will perform well if the computation time, τ , is less than both the dephasing and dissipation times, this is, below the limit (Γ + γ)τ < 1 (red dashed line in Figure 3A). dye systems that can be employed to map a one-qubit unitary transformations. Because these transformations can only be reliably realized if they are completed before bath induced relaxation, (Γ + γ) < 1/τ (red dotted line in Fig. 3) imposes an upper boundary on the allowable computation time. Furthermore, the transformation time, τ , depends on the strength of the coupling (e.g. τ = π/(2V 01 ) for the NOT gate). If the dyes can be placed no closer than some distance r min , and the orientation of the dyes is set to maximize the coupling from Eq. (2) (i.e., θ ij = π/2 and ϕ = 0), the coupling is bounded by the expression |V 01 | ≤ |µ A ||µ B |/4π 2 0 r 3 min . Using these considerations, we propose the following criterion promising excitonic circuits: where the choice of r min depends on the choice of supermolecular support structure for the dye pairs. For example, for a system of dyes embedded in DNA, the individual dyes cannot be placed closer than a DNA base-pair, so we have r min ≈ 3.4Å. For dye pairs in a polypeptide chain, r min is determined by the peptide size, which is typically 1.32Å. 34

Optimizing Circuit Geometry
We now consider optimizing the geometry of excitonic circuits to maximize their Fidelity.
Due to the flexibility of the constraints imposed by the unitary transformations many configurations can reproduce the same gate. However, these will generally differ in their computation time and therefore, their sensitivity to environmental noise. For the NOT gate, this time scale is entirely determined by the coupling between the dyes. For simplicity, we will restrict our attention to homodimers with identical dyes since this is the most likely method of achieving degenerate excited states. In this case, µ A = µ B and the coupling V 01 and, consequently, the fidelity will be a function of only the spatial arrangement of the molecular system. As pointed out in Section 2, the exciton geometry can be described by three parameters: the twist angle, θ ij , the center-to-center intermolecular distance, r ij , and the angle between the dyes and the distance vector, ϕ. In realistic systems, the bath contribution is expected to be dominated by the dephasing contribution. As such, we have restricted our attention to purely dephasing baths (i.e. Γ = 0) with a fixed slow dephasing rate γ * τ f = 0.8. The parameter τ f is calculated from the electronic coupling between a pair of nearly-orthogonal Cy3 dyes at base-pair distance, as in the previous section. This dephasing rate was selected to more clearly show the geometry dependence of the fidelity and is likely to be significantly higher in realistic systems.
We first consider a pair of dyes comprising a NOT gate that are displaced perpendicular to their dipole moments (i.e. ϕ = π/2), which sets the second therm in Eq. (2) to zero. Figure 4A shows the dependence of the fidelity of the NOT gate on the spatial terms, r ij and θ ij . Note that the intermolecular distance is presented as a ratio of the transition dipole moment magnitude, µ, inÅ/D units, with µ = 12D corresponding to a Cy3 homodimer.
Some interesting patterns in the behaviour of the fidelity should be highlighted from Figure 4A. We first note two regions where the fidelity is zero for all intermolecular distances: when θ ij = π/2 and 0, π. In the first case, the dyes are orthogonal to each other, leading to vanishing coupling for all r ij . The dye geometry is therefore incapable of satisfying the coupling constraints imposed by the NOT gate transformation. The second case corresponds to a dimer of parallel or antiparallel dyes. In this case, the the dyes can not be distinguished by the polarization addressing scheme. As a result, Eq. (8) results in a fidelity of zero when θ ij = 0, π, for all values of r ij . Overall, the dependence of the fidelity on the interchromophore angle, θ ij , demonstrates a trade-off between the cos θ ij and sin 2 θ ij terms, with a maximum fidelity at a critical point close to θ ij = π/2, as evident in Figure 4A.
The effect of the interchromophore distance, r ij , on the fidelity is contained in the term 1/r 3 ij in Eq. (2). As a result, we see that the coupling and the fidelity monotonically decrease with increasing distance between the dyes. In this case, the optimal geometry simply minimizes distance subject to experimental constraints.
A richer picture of geometry arrangement emerges when ϕ is allowed to vary. Figure 4B presents the behaviour of the fidelity when θ ij and ϕ are varied at a fixed ratio of 0.21e × r ij /µ = 1.29 and the same bath conditions as panel A. In this case, the coupling contains contributions form the second term in Eq. (2), allowing for V 01 = 0 when θ ij = π/2. As a result, the maximum fidelity can now be reached with orthogonal dyes as this circumvents the tradeoff between coupling and measurement specificity inherent to the ϕ = π/2 case treated above. Similarly to Fig. 4.A, a sharp line can be seen with rapidly decreasing fidelity where the dyes are uncoupled. Since the coupling depends on both angles, this is no longer a straight line and instead appears as the curved line in Fig. 4.B given by Figure 4: Fidelity of the NOT quantum gate for different spatial distributions of the dye-pair coordinate system. (A) Fidelity as a function of the dimensionless ratio of the inter-chromophore distance and transition dipole magnitude, 0.21e × r ij /µ, and angle, θ ij , when both dyes are orthogonal to the z axis, and (B) as a function of θ ij and ϕ, the angle between both dyes and the distance vector,r ij .
Optimizing the performance of the Hadamard gate is slightly more straightforward than the CNOT gate. From Table 1 we can note that the coupling of the Hadamard gate is constrained by the ratio ∆ 01 /V 01 = 2. As a result, τ and therefore the fidelity can be determined by comparing the relative energy between the dye molecules. It is therefore straightforward to show that, as long as the required coupling is achievable for a given pair of dyes, it is optimal to select a dye configuration with angle θ ij = π/2 in order to maximize the selectivity of polarization addressing. Figure 5 illustrates the dependence of the fidelity on the difference between the individual excitation energies of the dyes, ∆ 01 , for a fixed bath with Γ = 0 and γ * τ f = 0.8. We can see that the Fidelity increases rapidly with the energy difference, as the coupling also increases proportionally. However, we note that, as described by Eq. (9), the dipole moments and the achievable dye configurations provide a bound for the maximum coupling that can be implemented in the dye system. This limits the energy differences of dyes that can be used, capping the achievable Fidelity of a given circuit.

Conclusions
In this manuscript, we have proposed a general strategy for mapping specific unitary operations onto excitonic cricuits. We show that this strategy is limited in complexity to 12 dimensional or smaller quantum systems. However, these systems exhibit a manifold of possible excitonic circuits that are able to generate them. We show that these excess degrees of freedom can be exploited to facilitate experimental initialization and measurement and to mitigate the effect of environmental noise. Moreover, we identify a criterion for molecular parameters that identify dye-scaffolding systems that may be promising for quantum information applications.