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Dartmouth College Dartmouth College Dartmouth Digital Commons Dartmouth Digital Commons Open Dartmouth: Peer-reviewed articles by Dartmouth faculty Faculty Work 2020 Quantum Port-Hamiltonian Network Theory Quantum Port-Hamiltonian Network Theory Frederick I. Moxley III Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa Part of the Artificial Intelligence and Robotics Commons, Control Theory Commons, Dynamic Systems Commons, Non-linear Dynamics Commons, Quantum Physics Commons, and the Theory and Algorithms Commons

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Dartmouth College Dartmouth College

Dartmouth Digital Commons Dartmouth Digital Commons

Open Dartmouth: Peer-reviewed articles by Dartmouth faculty Faculty Work

2020

Quantum Port-Hamiltonian Network Theory Quantum Port-Hamiltonian Network Theory

Frederick I. Moxley III

Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa

Part of the Artificial Intelligence and Robotics Commons, Control Theory Commons, Dynamic Systems

Commons, Non-linear Dynamics Commons, Quantum Physics Commons, and the Theory and Algorithms

Commons

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Noname manuscript No.(will be inserted by the editor)

Quantum Port-Hamiltonian Network Theory

Universal Quantum Simulation with RLC Circuits

Frederick Ira Moxley III

Received: date / Accepted: date

Abstract Herein, we propose using Resistor-Inductor-Capacitor (RLC) cir-

cuits for achieving universal quantum simulators. This is accomplished by

explicitly presenting the required set of universal gates using RLC circuits

obtained from quantum network theory. Our proposal for universal quantum

simulators and quantum information processing systems is based on imple-

menting composite Dirac structures using RLC circuits. These Dirac struc-

tures can then be networked together such as to execute arbitrary quantum

algorithms using the interconnected universal RLC circuit gates, where vari-

able inductors simulate the quantum interactions. Owing to Digital-to-Analog

Conversion (DAC), our framework admits a fully-programmable architecture,

as the programmable input currents simulate the prepared quantum states for

any given algorithm. The resulting solution of the arbitrary quantum simula-

Frederick Ira Moxley III

Dartmouth College, Hanover, NH 03755 USA

E-mail: [email protected]

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2 Frederick Ira Moxley III

tion is given by the output voltage probability amplitudes of the interconnected

Dirac structures, converted to a bit array (i.e. bit map, bit set, bit string, or

bit vector) via Analog-to-Digital Conversion (ADC). As such, our construc-

tion is robust and stable, which can easily be manipulated at high speeds,

and stored (written) electronically, where the logical value 1 (high voltage) or

logic 0 (low voltage) is driven into the bit line of a Random-Access Memory

(RAM) memory cell. Due to the stability and high readout or transfer speeds

of RLC circuits, it is of particular interest to utilize these systems as universal

quantum simulators to realize the execution of arbitrary quantum algorithms.

Keywords Dirac structures · universal quantum simulators · probabilistic

Turing machines

1 Introduction

Following some earlier efforts [1,2], more recently there has been research ef-

forts to develop a Universal Quantum Simulator (UQS). UQSs can carry out

any possible computation, including simulating completely different models of

computation. Universal models of computation are able to simulate arbitrary

many-body physics phenomena, including reproducing the physics of arbitrar-

ily different many-body physics models [3]. Herein, we demonstrate that a

UQS can be attained by using analog components such as Resistor-Inductor-

Capacitor (RLC) circuits [4]. These RLC circuits can be assembled in a fashion

which emulates the behavior of a true quantum system [5,6] (e.g. integer fac-

torization [7,8]). This is useful for many applications, including but not limited

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Quantum Port-Hamiltonian Network Theory 3

to, quantum cryptography, and the efficient (i.e., polynomial-runtime) imple-

mentation of Universal models of computation. A faithful UQS must obey con-

servation laws (e.g. energy, charge, etc.) while they exhibit certain dynamical

features such as quantum discord [9], and entanglement [10]. In order to per-

form efficient quantum simulation of fermionic and frustrated systems (while

avoiding the exponential growth of statistical errors as the number of particles

increases [11]), the UQS must overcome the infamous sign problem [12]. Re-

cently, it has also been rigorously demonstrated in the peer-reviewed academic

literature that a fully-programmable UQS can be realized by using the inter-

connection of universal gates (e.g., coupled qubits [13,14], quantum walks [15,

16], two-level systems [17]). Following these and other motivations, herein we

present a novel method for constructing an analog UQS using RLC circuits.

These circuits scale linearly in the number of classical circuit elements neces-

sary for preparing and simulating an exponential number of quantum states

[18], and are essentially probabilistic Turing machines. This allows for the

implementation of universal models of computation, by providing the three el-

ementary gates necessary to do so [19]. These three universal gates are, namely,

the phase shift, the Hadamard, and the C-NOT gates. By programming the

individual mutual inductances and capacitances in the RLC framework, we

achieve the analog UQS. One can also achieve the UQS with solely phase-

shift and Hadamard gates, although in this way, the number of gates required

grows exponentially with the size of the problem being addressed. As such, the

phase-shift and Hadamard only approach is uninteresting, in the sense that it

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4 Frederick Ira Moxley III

does not outperform efficient classical computation. By introducing the RLC

C-NOT gate herein, the number of required gates is substantially reduced, as

compared to the number of gates a classical computer requires for an identi-

cal computational task. The RLC C-NOT gate enables the entanglement of

many-voltage states, such that a RLC quantum network performs as if it is

performing many different gate operations simultaneously, thereby function-

ing as a probabilistic Turing machine. Moreover, we address the coupling of

quantum channels (registers) [20], separated by an arbitrary distance such as

to mediate voltage supertransport, where the interaction between quantum

channels is mediated by the programmable mutual inductances [21]. In doing

so, we successfully analog-simulate universal quantum walks, thus allowing

for programmable quantum computing operations. By considering a quan-

tum network with RLC circuits, an arbitrary universal quantum walk can be

programmed. Another important component of any quantum algorithmic im-

plementation is to consider the initial state preparation, and the I/O with

respect to that initial state. The RLC analog UQS initial state (I) is prepared

(programmed) via Digital-to-Analog Conversion (DAC), and the output state

(O) undergoes Analog-to-Digital Conversion (ADC) such that the voltage is

stored (cf. quantum measurement or observation) in Random-Access Memory

(RAM) cells. This is particularly useful for the case in which the voltage output

contains a small number of bits (e.g. computing a problem where the output is

binary). the voltage I/O conversion process is described in detail by present-

ing a novel quantum Port-Hamiltonian theory, which portrays the voltage I/O

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Quantum Port-Hamiltonian Network Theory 5

conversion process as an interconnection of Dirac structures (i.e., rigorously

and well-defined mathematical objects). In this case, the Dirac structures rep-

resent the analog UQS-RLC gates in terms of effort and flow variables (voltages

and currents, respectively).

The purpose of this article is to investigate the prospects of using RLC

circuits to construct universal quantum simulators, and to formulate a the-

oretical description of its operation using time-dependent composite Dirac

structures. In §2 we introduce the idea of using RLC circuits to prepare quan-

tum analog voltage states, and present analytical solutions for the bright and

dark states. We then obtain the quantized energy relation from the electro-

magnetic radiation of the RLC circuit, and derive the RLC Bell-states from

the Shockley diode equation. In §3 we present the quantum port-Hamiltonian

framework, and illustrate its application using RLC circuits. These RLC cir-

cuits are then used to construct the primitive gateset for universal quantum

simulation, i.e., the phase-shift, Hadamard, and CNOT gates, and are synthe-

sized with the composite Dirac structures of the quantum port-Hamiltonian

framework. Finally, a Schrodinger equation for the universal RLC quantum

circuit is obtained, and concluding remarks are made in §4.

1.1 Preliminaries

Definition 1 Let P1 ∈ Cn×n be invertible and self-adjoint, let P0 ∈ Cn×n be

skew-adjoint, i.e., P †0 = −P0, and let H ∈ L∞([j = 0, . . . , N ];Cn×n), where

j ∈ N such that H†j = Hj, mI ≤ Hj ≤ MI for a.e. j ∈ [0, N ] and constants

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6 Frederick Ira Moxley III

m,M > 0 independent of j [22]. We equip the Hilbert space X := L2([j =

0, . . . , N ];Cn) with the discrete inner product

〈ψ,ϕ〉X =1

4〈ϕ0|H0|ψ0〉+

1

2

N−1∑j=1

〈ϕj |Hj |ψj〉+1

4〈ϕN |HN |ψN 〉 . (1)

Then the linear, first-order Schrodinger equation

i~ |ψ(t)〉 =P1

2

∑j

(Hj+1 −Hj−1

)|ψj(t)〉

+P1

2

∑j

Hj(|ψj+1(t)〉 − |ψj−1(t)〉

)+ P0

∑j

Hj |ψj(t)〉 (2)

is a quantum port-Hamiltonian system, where ~ is the reduced Planck constant,

the imaginary number i =√−1, and the instantaneous state of the quantum

system at time t is

|ψ(t)〉 =∑j

αj(t) |j〉 (3)

where the complex number αj(t) is

αj(t) = 〈j|ψ(t)〉 , and 〈j′|j〉 = δjj′ . (4)

Definition 2 Consider a finite-dimensional linear space F ∈ Ck with E = F†.

A subspace D ⊂ F ⊗ E is a Dirac Structure if [23]:

1. 〈e|f〉 = 0, ∀ (f, e) ∈ D,

2. dim D = dim F .

Definition 3 Consider a state space manifold χ and a Hamiltonian H :

χ → C, defining energy-storage [24], e.g. current I or voltage V . A quantum

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Quantum Port-Hamiltonian Network Theory 7

port-Hamiltonian system on χ is defined by the Dirac structure

D ⊂ Tχχ⊗ T †χχ⊗FP ⊗ EP , (5)

having energy-storing port (fS , eS) ∈ Tχχ⊗T †χχ, and an external structure P,

e.g. source voltage or output current, such that

P ⊂ FP ⊗ EP , (6)

corresponding to an external port (fP , eP) ∈ FP ⊗EP . The temporal dynamics

of the quantum system are then specified by{αj(t) =

∂πj〈H〉 ,−πj(t) =

∂αj〈H〉 , fP(t), eP(t)

}∈ D

(αj(t)

), t ∈ R;

(7)

where αj are the generalized coordinates, and πj are the conjugate momenta.

Remark 1.1 The property D = D⊥ can be regarded as a generalization of

Tellegen’s Theorem from circuit theory, since it describes a constraint between

two different realizations of the port variables (Ibid.), in contrast to property

1 of Definition 2.

Remark 1.2 In the infinite-dimensional case (Ibid.), the property D = D⊥ will

be taken as the definition of an infinite-dimensional Dirac structure.

Remark 1.3 The 2D Heisenberg and XY models with variable coupling strengths

(c.f. mutual inductances) are a class of two-qubit interactions that can simulate

any stoquastic Hamiltonian, i.e. any Hamiltonian whose off-diagonal entries

in the standard basis are nonpositive [3]. As such, these models are universal

simulators, i.e. the class of Hamiltonians believed not to suffer from the sign

problem in numerical Monte Carlo simulations [12].

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8 Frederick Ira Moxley III

2 The RLC Quantum Channel

2.1 Quantum State Preparation

We begin by adapting some classical theory of the Operational Transconduc-

tance Amplifier (OTA) [25], for purposes of quantum state preparation with

universal RLC circuit design, as seen in Figure 2.1.3. The transconducting

gain gm is proportional to the external dc bias current Iext, where the pro-

portionality constant ~ is dependent upon temperature, device geometry, and

the process [26]. Furthermore, we assume the input and output impedances Z

have ideal values of infinity, i.e. Av = Rin =∞ and Rout = 0. As such,

gm = ~Iext. (8)

Controllability of the gain gm, and hence the quantum state preparation can

be obtained by programming Iext using Digital-to-Analog Conversion (DAC)

techniques [27]. Furthermore, the output current of an OTA is proportional to

the input signal voltage [28], such that

Iout(t) = gm

(V +in (t)− V −in (t)

), (9)

where the complex-valued voltages are

V +j (t) = |V +| exp(iω+

j t), (10a)

V −j (t) = |V −| exp(iω−j t); (10b)

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Quantum Port-Hamiltonian Network Theory 9

(DAC) |ψ0〉 D(t) =∑

j U(t) |ψj(0)〉 (ADC)

√22 dj

e.g. (S ∨D)-RAM

dz = dj = 1

time

+

gm(V+ |1〉−V − |0〉)

V − |0〉

V + |1〉

LjLj−1 Lj+1

Q

BL

Q†BL†

WL

(a) RLC Quantum Channel

eV = −fI

fV = eI

eP = V = ϕfP = I = ϕ/L

−〈eV |fV 〉 = 〈fI |eI〉

〈eV |fV 〉+ 〈eI |fI〉+ 〈eP |fP〉 = 0 7→ H ≤ 〈eP |fP〉

χ(I) D

P

χ(V )eI = fV

fI = −eV

(b) Quantum Port-Hamiltonian System

Fig. 1 (a) RLC circuit equivalent of the quantum channel. We work in natural units, where

R = ~ = 1, L = C = t = 1/eV, ωj = 2π eV, and I = 1eV. Digital-to-Analog Conversion

(DAC) is used to prepare the analog quantum state, which then propagates along the z-

direction. The time-dependent Dirac structure D(t) then performs a unitary evolution of

the prepared quantum state. The current across the resistor(s) is IR(t) = |Ij | sin(ϕj), the

current across the capacitor(s) is IC(t) = iωjCj |Vj | cos(ϕj), and ϕj(t) = ωjt is the RLC

quantum phase. The output of the Dirac structure then undergoes Analog-to-Digital Con-

version (ADC), such that the quantum information is stored (cf. measurement) in Random-

Access Memory (RAM) cell(s). (b) Quantum Port-Hamiltonian System representation of

the RLC quantum channel (a), i.e. the port interconnection of current I and voltage V , the

external port P, where χ(I) is the current storage state, χ(V ) is the voltage storage state, the

Dirac structure D links the storage ports (flows {fI , fV } and efforts {eI , eV }, respectively),

with the external port (flow fP and effort eP ). The electric power-balance (conservation)

equation is H = i~∂tt, such that the total power is equal to zero (color online).

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10 Frederick Ira Moxley III

and ω±j t are the angular frequencies. When an ideal resistor RL = Z0 is

connected to the output of an OTA, a simple voltage amplifier is obtained:

Vout

V +in − V

−in

= gmRL. (11)

Next we write the complex-valued current as

I = |Ij | exp[i(ωIt− φ)]. (12)

With Eqs. (10a)-(10b) and after DAC, the j-th output current (i.e. prepared

analog quantum superposition state) of an OTA in the digital basis is:

Ioutj (t) = gm

(|V +j | exp[iω+

j t] |1〉 − |V−j | exp[iω−j t] |0〉

), (13)

where the logical value(s) 1 represents high voltage(s), and the logical value(s)

0 represents the low voltage(s). By combining Eqs. (12)-(13), we then obtain

the j-th RLC circuit site current with continuous (time-dependent) phases

|Ioutj (t)| = gmexp[i(ωIt− φ)]

(|V +j | exp[iω+

j t] |1〉 − |V−j | exp[iω−j t] |0〉

). (14)

For clarification purposes cf. qubits, here it should be pointed out that the

quantum superposition state(s) prepared by the OTA(s) is |ψ(t)〉 = |Ioutj (t)〉,

such that

|ψ(t)〉+ =√I+j exp[i(ω+

I t− φ+)] |1〉 , (15a)

|ψ(t)〉− =√I−j exp[i(ω−I t− φ

−)] |0〉 ; (15b)

and the superposition state |ψ(t)〉 = |ψ(t)〉+±|ψ(t)〉−. Hence, from Ohm’s law,

the output voltage amplifier has prepared for ADC the OTA output voltage

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Quantum Port-Hamiltonian Network Theory 11

state equation

Vout(t) = RLIoutj (t). (16)

As seen in Figure 1(a), in natural units, where R = ~ = 1, L = C = t = 1/eV,

ωj = 2π eV, and I = 1eV, by taking C = 2/eV we finally obtain the coupled

Schrodinger equations governing the electrodynamics of the RLC quantum

channel [29,30,31]

i~Vj(t) |ψ(t)〉+ =1

2m

[Ij−1(t)− Ij+1(t)

]|ψ(t)〉+ − |Ij | sin

(2π

ϕ0ϕj(t)

)|ψ(t)〉− ,

i~Vj(t) |ψ(t)〉− =1

2m

[Ij−1(t)− Ij+1(t)

]|ψ(t)〉− − |Ij | sin

(2π

ϕ0ϕj(t)

)|ψ(t)〉+ ;

(17)

where Ij(t) is the current, Vj(t) is the voltage, and ϕj(t) is the RLC quantum

phase, i.e.

Ij(t) = − 1

2Lj[Vj+1(t)− Vj−1(t)],

Vj(t) = − 1

2Cj

[Ij+1(t)− Ij−1(t) + 2|Ij | sin

(2π

ϕ0ϕj(t)

)],

ϕ(t) = Vj(t). (18)

Furthermore, here it should be pointed out that the position observable x =

z = j∆z = j, and the momentum observable p = −i~∂1/2/∂z1/2, thereby

satisfying the Heisenberg uncertainty principle [32]

〈x2〉 〈p2〉 ≥ ~4. (19)

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12 Frederick Ira Moxley III

2.1.1 Bright States

The first solution of Eq. (18) is a bright soliton propagation along the RLC

quantum channel as seen in Figure 1(a). As such, the analytical solution is

ϕj(t) = 2ϕ0

πatan

[exp

( j − vt

λ√

1− v2

c2

)](20a)

Vj(t) =ϕ0

2πϕj(t)

= −ϕ0ω

2v√1− v2

c2

sech[ j − vt

λ√

1− v2

c2

](20b)

Ij(t) = − ϕ0

4πLj

(ϕj+1(t)− ϕj−1(t)

)= − ϕ0

2πLjλ

2√1− v2

c2

sech[ j − vt

λ√

1− v2

c2

](20c)

2.1.2 Dark States

The second solution of Eq. (18) is a dark soliton propagation along the RLC

quantum channel as seen in Figure 1(a). As such, the analytical solution is

ϕj(t) = 2ϕ0

πatan

[exp

( j − vt

λ√

1− v2

c2

)](21a)

Vj(t) =ϕ0

2πϕj(t)

= −ϕ0ω

2v√1− v2

c2

tanh[ j − vt

λ√

1− v2

c2

](21b)

Ij(t) = − ϕ0

4πLj

(ϕj+1(t)− ϕj−1(t)

)= − ϕ0

2πLjλ

2√1− v2

c2

tanh[ j − vt

λ√

1− v2

c2

](21c)

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Quantum Port-Hamiltonian Network Theory 13

2.1.3 Electromagnetic Radiation of the RLC Quantum Circuit

The quantization of the electromagnetic radiation in the quantum RLC circuit

is acheived by considering πj′ and qj as formally equivalent to the momentum

and coordinate of a quantum mechanical harmonic oscillator. Therefore, we

take the commutator relations connecting the quantum RLC circuit dynamical

variables as

[πj , πj′ ] = [qj , qj′ ] = 0, [qj , πj′ ] = i~δj,j′ . (22)

We then define the creation operator a†j(t) and the annihilation operator aj(t)

by using Eq. (58) such that

a†j(t) =

√1

2~ωj[ωjqj(t)− iπj(t)]

=

√1

2~ωj

∫ tf

t0

[ωj

∂πj〈H(αj , πj , t

′)〉+ i∂

∂αj〈H(αj , πj , t

′)〉]dt′

=

√1

2~ωj

∫ tf

t0

[ ∂∂t

(ωjαj(t

′)− iπj(t′))]dt′

=

√1

2~ωj

∫ tf

t0

[ ∂∂t

(ωjαj(t

′) + ~αj(t′))]dt′

=

√1

2~ωj

[ωjαj(t) + ~αj(t)

], (23a)

aj(t) =

√1

2~ωj[ωjqj(t) + iπj(t)]

=

√1

2~ωj

∫ tf

t0

[ωj

∂πj〈H(αj , πj , t

′)〉 − i ∂

∂αj〈H(αj , πj , t

′)〉]dt′

=

√1

2~ωj

∫ tf

t0

[ ∂∂t

(ωjαj(t

′) + iπj(t′))]dt′

=

√1

2~ωj

∫ tf

t0

[ ∂∂t

(ωjαj(t

′)− ~αj(t′))]dt′

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14 Frederick Ira Moxley III

=

√1

2~ωj

[ωjαj(t)− ~αj(t)

]. (23b)

The formal analogy between the operators a†j , and aj and their counterparts in

the case of harmonic oscillators show that, quantum mechanically, a stationary

state of the total radiation field can be characterized by an eigenfunction

Φ, which is a product of the eigenfunctions of the individual Hamiltonians

~ωj(a†j aj + 1/2)

Φ = un1un2· · · =

∞∏j=1

unj, (24)

where unjare a complete orthonormal set of basis functions, and

a†junj =√nj + 1unj+1, ajunj =

√njunj−1, a†j ajunj = njunj . (25)

The expectation value of the number operator a†j aj is then

〈Φ|a†j aj |Φ〉 = 〈nj |a†j aj |nj〉 = nj , (26)

and is equal to the number of quanta nj in the j-th mode of the quantum

channel. More specifically, a quantum channel of length Lj along the axis

of mode volume V = dkxdkydkz = (2π/Lj)3 in k space with electric and

magnetic field vectors pointing in the y- and x- directions, respectively, satisfy

the Hamiltonian

H =1

2

∫V

∑j

(µHj ·Hj + εEj ·Ej)dv. (27)

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Quantum Port-Hamiltonian Network Theory 15

As such,

E(x, t) = i∑j

√~ωjV ε

[a†j(t)− aj(t)] sin(j · kj)x

= i

N∑j=1

√~ωjV ε

[a†j(t)− aj(t)] sin(j · π

N〈nj |a†j aj |nj〉

)x

= i~√

2

V ε

N∑j=1

αj(t) sin(j · π

2~ωjN[ωjα

2j (t)− ~2α2

j (t)])x, (28a)

H(x, t) =∑j

√~ωjV µ

[a†j(t)− aj(t)] cos(j · kj)y

=∑j

√~ωjV µ

[a†j(t)− aj(t)] cos(j · π

N〈nj |a†j aj |nj〉

)y

= ~√

2

V µ

N∑j=1

αj(t) cos(j · π

2~ωjN[ωjα

2j (t)− ~2α2

j (t)])y; (28b)

where the wavevector kj = πnj/Lj . Upon inserting those quantized electric

and magnetic fields into the Hamiltonian Eq. (27) using Eqs. (23)-(23), we ob-

tain the time-dependent quantized energy relation for theN -quantum channels

H(t) =~cn

N∑j=1

βj

(a†j(t)aj(t) +

1

2

), (29)

where β2j ∼ ω2

j ε/c2, n =

√ε, the creation operators a†j(t), and the annihilation

operators aj(t) are given by Eq. (23), and Eq. (23), respectively.

2.2 Analog Bell-State Preparation

In order to describe the Bell-State preparation as seen in Figure 2.1.3 [33], we

invoke the Schockley diode equation for describing the mixing of the voltage

state equations [34], i.e.

I = IS

[exp

( VDnVT

)− 1]

(30)

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16 Frederick Ira Moxley III

+

R· · · ,inL|· · ·〉in g···m(V+· · · ,in − V

−· · · ,in)

V +· · · ,in

V −· · · ,in

V· · · ,in

+

Rd,inL|d〉in gdm(V

+d,in − V

−d,in)

V +d,in

V −d,in

Vd,in

+

Rc,inL|c〉in gcm(V

+c,in − V

−c,in)

V +c,in

V −c,in

Vc,in

+

Rb,inL|b〉in gbm(V

+b,in − V

−b,in)

V +b,in

V −b,in

Vb,in

+

Ra,inL|a〉in gam(V

+a,in − V

−a,in)

V +a,in

V −a,in

Va,in

+

R· · · ,outL

+

Rd,outL

+

Rc,outL

+

Rb,outL

+

Ra,outL

Da||Db||Dc||Dd||D···

g···m(V+· · · ,out − V −· · · ,out)

gdm(V+d,out − V

−d,out)

gcm(V+c,out − V −c,out)

gbm(V+b,out − V

−b,out)

gam(V+a,out − V −a,out) |a〉out ⊗ |b〉out

|c〉out ⊗ |d〉out

|· · ·〉out ⊗ |· · ·〉out

Fig. 2 Controllability of the gain(s) gm, and hence the quantum state preparation can be

directly programmed using Digital-to-Analog Conversion (DAC) techniques. The composite

universal Dirac structure Da||Db||Dc||Dd||D··· intakes the prepared quantum states, and

outputs an analog voltage representing the quantum output states, where the logical value 1

(high voltage) or logic 0 (low voltage) is driven into the bit line of a Random-Access Memory

(RAM) memory cell via Analog-to-Digital Conversion (ADC) (color online).

where I is the diode current, IS is the reverse bias saturation current (scale

current), VD is the voltage across the diode, VT is the thermal voltage, and n is

the ideality factor, also known as the quality factor, or the emission coefficient.

Upon Taylor expanding the exponential term, and neglecting the constant

coefficients in the Schockley diode equation, the output voltage will have the

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Quantum Port-Hamiltonian Network Theory 17

form

|a〉out ⊗ |b〉out = Ra,outL |Ia,outm 〉+Rb,out

L |Ib,outm 〉

+1

2

[(Ra,out

L )2 |Ia,outm , Ia,outm 〉+ 2Ra,outL Rb,out

L |Ia,outm , Ib,outm 〉

+ (Rb,outL )2 |Ib,outm , Ib,outm 〉

], (31)

and similarly for |c〉out⊗ |d〉out. As such, we obtain the Bell states as depicted

in Figure 2.1.3:

|Φ±〉 =1√2

(|a〉out ⊗ |c〉out ± |b〉out ⊗ |d〉out), (32a)

|Ψ±〉 =1√2

(|a〉out ⊗ |d〉out ± |b〉out ⊗ |c〉out). (32b)

3 Quantum Port-Hamiltonian Networks

Port-Hamiltonian systems have been studied extensively for the case of clas-

sical RLC-circuits [35,36]. Herein we aim to extend the port-Hamiltonian

framework to RLC quantum networks [4], e.g. Figure 1(a), for the purpose

of developing universal analog quantum computers [37]. The standard way of

modeling the system in Figure 1(a) is to start with the configuration of the

charge (storage state) Q ∈ χ, and to write down the classical Hamiltonian of

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18 Frederick Ira Moxley III

the RLC circuit, i.e.

H(Q,ϕ) =1

2m(LjIj)

2 +1

2kQ2

j

=1

2

∑j

[Lj

( ∂∂tQj

)2+

1

Cj

( ∂∂zQj

)2]=

1

2

∑j

[Lj

(− ∂

∂zIj

)2+

1

Cj

( ∂∂zQj

)2]

=1

2

∑j

[Lj

(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2 ](33)

since Q = −∂zI, the energy stored in an inductor T = (LI)2/2m = ϕ2(t)/2m,

and the total electric potential energy stored in a capacitor is given by U =

kQ2/2, where C is the capacitance, V is the electric potential difference, and

Q is the charge stored in the capacitor, i.e. m = L and k = 1/C. Quantum

port-Hamiltonian networks can be regarded as treating the kinetic and poten-

tial energies as interconnected subsystems, both of which store energy. Now

suppose we have a set of basis states {|n〉} that are discrete, and orthonormal,

i.e.

〈n′|n〉 = δnn′ . (34)

The instantaneous state of the RLC quantum circuit at time t can be expanded

in terms of these basis states, viz.

|ψ(t)〉 =∑n

αn(t) |n〉 , (35)

where

αn(t) = 〈n|ψ(t)〉 . (36)

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Quantum Port-Hamiltonian Network Theory 19

From the expansion of the state in terms of basis states, the expectation value

of the Hamiltonian is

〈H(t)〉 = 〈ψ(t)|H|ψ(t)〉

=1

2

∑j

∑nn′

αn′(t)αn(t)

⟨n′

∣∣∣∣∣Lj(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2∣∣∣∣∣n⟩.

(37)

With the generalized canonical coordinate q = αn(t) = Q, and the conjugate

momenta πn(t) = i~αn(t) = ϕ, it can be seen that

∂αn′〈H(t)〉 = 〈ψ(t)|H|ψ(t)〉

=1

2

∑j

∑n

αn(t)

⟨n′

∣∣∣∣∣Lj(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2∣∣∣∣∣n⟩

=1

2

∑j

⟨n′

∣∣∣∣∣Lj(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2∣∣∣∣∣ψ(t)

⟩. (38)

It is well-known that a state vector |ψ(t)〉 evolves according to the Schrodinger

equation:

i~∂

∂t|ψ(t)〉 = H |ψ(t)〉 . (39)

As such, in addition to using the orthonormality of the basis states, Eq. (38)

can be written

∂αn′〈H(t)〉 = i~

∂tαn′ . (40)

Similarly,

∂αn〈H(t)〉 = −i~ ∂

∂tαn. (41)

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20 Frederick Ira Moxley III

Hence, we obtain the quantum mechanical equations of motion for the total

system as seen in Figure 1(a) as

∂t

αnπn

=

0 1

−1 0

∂∂αn

⟨H(αn, πn)

⟩∂∂πn

⟨H(αn, πn)

⟩ . (42)

Moreover, the input-state-output quantum port-Hamiltonian RLC circuit as

seen in Figure 1(b) with (input) control voltage u := eP = Vin, and output

current y := fP = Iout is written

∂t

αnπn

=

0 1

−1 0

∂∂αn

⟨H(αn, πn)

⟩∂∂πn

⟨H(αn, πn)

⟩+

1

0

Vin,

Iout =

[1 0

] ∂∂αn

⟨H(αn, πn)

⟩∂∂πn

⟨H(αn, πn)

⟩ ; (43a)

where the skew-Hermitian adjoint structure matrix J , i.e. J† = −J is

J =

0 1

−1 0

. (44)

This leads to the system of equations for the current:

Current :

Q = αn(t) = −fI ,

eI = ∂∂αn

⟨H(αn, πn)

⟩;

(45)

where the flow −fI ∈ FI denotes the current, and the effort eI ∈ EI is the

voltage. The reason for the minus sign in front of fI is that we want the product

fIeI to be the incoming power with respect to the port interconnection as seen

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Quantum Port-Hamiltonian Network Theory 21

in Figure 1(b). We obtain similar equations for the voltage:

Voltage :

ϕ = πn = −fV ,

eV = ∂∂πn

⟨H(αn, πn)

⟩.

(46)

We then couple the current and the voltage subsystems to each other through

the interconnection element as detailed in Figure 1(b), using the commutation

relations

[αn(t), αn′(t)] = [πn(t), πn′(t)] = 0, [αn(t), πn′(t)] = i~δnn′ ; (47)

viz.,

Interconnection :

−fI = eV ,

fV = eI .

(48)

Here it should be pointed out that the RLC quantum channel interconnection

can be generalized for an arbitrary number of Dirac structures [22].

3.1 Mutually Inducting Quantum RLC Circuits

We now consider the case of coupled quantum channels [38,39], represented as

RLC circuits. Our way of modeling the system is to start with the configuration

of the charge (storage state) Q ∈ χ, and to write down the Hamiltonian of the

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22 Frederick Ira Moxley III

dz = dj = 1

Mj−1,k Mj,k Mj+1,k

√22 dj

√22 dj

dz = dj = 1

time

+

+

gm∆Va

V −a |0〉

V +a |1〉

gm∆VbV −b |0〉

V +b |1〉

(a) Interconnected RLC Circuit Equivalent

eP,a = Va = ϕa

eP,b = Vb = ϕb

eV,a = −fI,a

eV,b = −fI,b

fV,a = eI,a

fV,b = eI,b

fP,b = Ib = ϕb/L

fP,a = Ia = ϕa/L

〈eV,a|fV,a〉+ 〈eI,a|fI,a〉+ 〈eP,a|fP,a〉 = 0 7→ Ha ≤ 〈eP,a|fP,a〉

〈eV,b|fV,b〉+ 〈eI,b|fI,b〉+ 〈eP,b|fP,b〉 = 0 7→ Hb ≤ 〈eP,b|fP,b〉

fa

fb

ea

eb

χ(Ia)

χ(Ib)

Da

Db

Pa

Pb

χ(Va)

χ(Vb)

eI,a = fV,a

fI,a = −eV,a

eI,b = fV,b

fI,b = −eV,b

(b) Composition of Dirac Structures

Fig. 3 (a) RLC circuit equivalent of the coupled quantum channels propagating along the

z-direction, Eq. (18), where the current across the resistor(s) is IR(t) = |Ij | sin(ϕj), the

current across the capacitor(s) is IC(t) = iωjCj |Vj | cos(ϕj), and ϕj(t) = ωjt is the RLC

quantum phase [31] and (b) Interconnected quantum port-Hamiltonian system representing

the RLC coupled quantum channels, i.e. the composition of Dirac structure Da and Dirac

structure Db (color online).

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Quantum Port-Hamiltonian Network Theory 23

system, i.e.

H(Q,ϕ) =1

2

∑j

[Lj

( ∂∂tQj

)2+

1

Cj

( ∂∂zQj

)2]±M

∑jk

∂tQj

∂tQk(1− δjk)

=1

2

∑j

[Lj

(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2 ]

±M∑jk

(Ij−1 − Ij+1

2

)(Ik−1 − Ik+1

2

)(1− δjk) (49)

since Q = −∂zI, and the mutual inductance between the quantum channels

propagating in the z-direction is given by the nonlinear coefficient of M . Now

suppose we have two sets of basis states {|n〉} and {|m〉} that are discrete,

and orthonormal, i.e. one set of basis states for each RLC quantum channel

〈n′|n〉 = δnn′ and 〈m′|m〉 = δmm′ . (50)

The instantaneous state of the coupled RLC quantum channels a and b at time

t can be expanded in terms of a quantum superposition these basis states, viz.

|ψ(t)〉 =∑n

αn(t) |n〉+∑m

βm(t) |m〉 , (51)

where

αn(t) = 〈n|ψa(t)〉 and βm(t) = 〈m|ψb(t)〉 . (52)

From the expansion of the state in terms of basis states, the expectation value

of the Hamiltonian Eq. (49) is

〈H(t)〉 = 〈ψ(t)|H|ψ(t)〉

=∑nn′

〈n′|αn′(t)αn(t)H|n〉+∑mn′

〈n′|αn′(t)βm(t)H|m〉

+∑m′n

〈m′|βm′(t)αn(t)H|n〉+∑m′m

〈m′|βm′(t)βm(t)H|m〉 . (53)

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24 Frederick Ira Moxley III

With the generalized canonical coordinates qa = αn(t) = Qa, qb = βm(t) = Qb,

and the conjugate momenta πan(t) = i~αn(t) = ϕa, πbm(t) = i~βm(t) = ϕb, it

can be seen that

∂αn′〈H(t)〉 = 〈n′|Hαn(t)|n〉+ 〈n′|Hβm(t)|m〉

= 〈n′|H|ψa(t)〉+ 〈n′|H|ψb(t)〉 , (54a)

∂βm′〈H(t)〉 = 〈m′|Hαn(t)|n〉+ 〈m′|Hβm(t)|m〉

= 〈m′|H|ψa(t)〉+ 〈m′|H|ψb(t)〉 ; (54b)

∂αn〈H(t)〉 = 〈n′|αn′(t)H|n〉+ 〈m′|βm′(t)H|n〉

= 〈ψa(t)|H|n〉+ 〈ψb(t)|H|n〉 , (54c)

∂βm〈H(t)〉 = 〈n′|αn′(t)H|m〉+ 〈m′|βm′(t)H|m〉

= 〈ψa(t)|H|m〉+ 〈ψb(t)|H|m〉 . (54d)

For the case of coupled RLC quantum channels, the state vector |ψ(t)〉 evolves

according to the coupled nonlinear Schrodinger equations:

i~∂

∂t|ψa(t)〉 =

1

2

∑j

[Lj

(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2 ]|ψa(t)〉

±M∑jk

(Ij−1 − Ij+1

2

)(Ik−1 − Ik+1

2

)(1− δjk) |ψb(t)〉 , (55a)

i~∂

∂t|ψb(t)〉 =

1

2

∑j

[Lj

(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2 ]|ψb(t)〉

±M∑jk

(Ij−1 − Ij+1

2

)(Ik−1 − Ik+1

2

)(1− δjk) |ψa(t)〉 ; (55b)

where the +M corresponds to a bright soliton solution, and the −M corre-

sponds to a dark soliton solution. As such, in addition to using the orthonor-

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Quantum Port-Hamiltonian Network Theory 25

mality of the basis states, Eqs. (54a)-(54d) can be rewritten

∂αn′〈H(t)〉 = i~

∂tαn′ ,

∂βm′〈H(t)〉 = i~

∂tβm′ . (56)

Similarly,

∂αn〈H(t)〉 = −i~ ∂

∂tαn,

∂βm〈H(t)〉 = −i~ ∂

∂tβm. (57)

Hence, we obtain the quantum mechanical equations of motion for the coupled

RLC quantum channels as

∂t

αn

πan

βm

πbm

=

0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0

∂∂αn

⟨H(αn, π

an, βm, π

bm)⟩

∂∂πb

n

⟨H(αn, π

an, βm, π

bm)⟩

∂∂βm

⟨H(αn, π

an, βm, π

bm)⟩

∂∂πb

m

⟨H(αn, π

an, βm, π

bm)⟩

. (58)

This leads to the coupled system of equations for the currents:

Currenta :

Qa = αn(t) = −fI,a,

eI,a = ∂∂αn

⟨H(αn, π

an, βm, π

bm)⟩,

(59a)

Currentb :

Qb = βm(t) = −fI,b,

eI,b = ∂∂βm

⟨H(αn, π

an, βm, π

bm)⟩

;

(59b)

where the flows −fI,a ∈ Fa, and −fI,b ∈ Fb denote the currents, and the

efforts eI,a ∈ Ea, and eI,b ∈ Eb, are the voltages. The reason for the minus

signs in front of the fI ’s is that we want the products fI,aeI,a and fI,beI,b to

be the incoming powers with respect to the port interconnections. We then

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26 Frederick Ira Moxley III

obtain similar equations for the voltages:

Voltagea :

ϕa = πan = −fV,a,

eV,a = ∂∂πa

n

⟨H(αn, π

an, βm, π

bm)⟩,

(60a)

Voltageb :

ϕb = πbm = −fV,b,

eV,b = ∂∂πb

m

⟨H(αn, π

an, βm, π

bm)⟩.

(60b)

We then couple the current and the voltage subsystems to each other through

the interconnection element using the commutation relations

[αn(t), αn′(t)] = [πan(t), πan′(t)] = 0, [αn(t), πan′(t)] = i~δnn′ , (61a)

[βm(t), βm′(t)] = [πbm(t), πbm′(t)] = 0, [βm(t), πbm′(t)] = i~δnn′ ; (61b)

viz.,

Interconnection :

−fI,a = eV,a,

fV,a = eI,a;

−fI,b = eV,b,

fV,b = eI,b.

(62)

3.2 Composite Dirac Structures

We now consider a Dirac structure Da on a product space Fa ⊗ Fa⊗b of two

linear spaces Fa and Fa⊗b, and another Dirac structure Db on a product

space Fa⊗b ⊗ Fb, where Fb is also a linear space. The linear space Fa⊗b is

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Quantum Port-Hamiltonian Network Theory 27

V −a |0〉

V +a |1〉

δ

~ω ~ω(1−∆)

~ω(1 + δ)

(a) Energy-Level Diagram

dz = dj = 1

Lj−1 MjLj+1

~ω(1 + δ −∆)

√22 dj

√22 dj

dz = dj = 1

time

+

+

gm∆Va

V −a |0〉

V +a |1〉

gm∆VbV −b |0〉

V +b |1〉

(b) RLC Quantum Phase-Shift Gate

Fig. 4 RLC circuit equivalent of the Phase-Shift Gate in the dual-rail encoding [41],

where the soliton propagates in the z-direction, as described by Eq. (18). The cur-

rent across the resistor(s) is IR(t) = |Ij | sin(ϕj), the current across the capacitor(s) is

IC(t) = iωjCj |Vj | cos(ϕj), and ϕj(t) = ωjt is the RLC quantum phase, and M is the mu-

tual inductance. The phase-shift is acheived by tuning the inductances δL and ∆L, such

that |a〉out = exp(iΦ) |a〉in. Not drawn to scale (color online).

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28 Frederick Ira Moxley III

the space of shared flow variables, i.e. {fa, fb}, and F†a⊗b is the the space

of shared effort variables, i.e. {ea, eb} as shown in Figure 3(b). In order to

interconnect Da with Db, the sign convention for the power flow corresponding

to (fa⊗b, ea⊗b) ∈ Fa⊗b ⊗ F†a⊗b must first be addressed. Taking 〈e|f〉 as the

incoming power, then since

(fI,a, eI,a, fV,a, eV,a, fP,a, eP,a, fa, ea) ∈ Da

⊂ Tχ,I,aχI,a ⊗ T †χ,I,aχI,a ⊗ Tχ,V,aχV,a ⊗ T†χ,V,a

χV,a

⊗FP,a ⊗F†P,a ⊗Fa⊗b ⊗F†a⊗b (63)

we have the power coming into Da denoted as 〈ea|fa〉 owing to the flow and

effort variables (fa, ea) ∈ Fa⊗b ⊗F†a⊗b. Similarly,

(fb, eb, fI,b, eI,b, fV,b, eV,b, fP,b, eP,b) ∈ Db

⊂ Fa⊗b ⊗F†a⊗b ⊗ Tχ,I,bχI,b ⊗ T†χ,I,b

χI,b

⊗ Tχ,V,bχV,b ⊗ T †χ,V,bχV,b ⊗FP,b ⊗F†P,b (64)

the term 〈eb|fb〉 denotes the power coming into Db. Here, it is obvious that

the power coming into Da owing to the power variables in Fa⊗b⊗Ea⊗b should

equal the outgoing power into Db. As such, we introduce the interconnection

constraints:

fa = −fb ∈ Fa⊗b, ea = eb ∈ (F†a⊗b = Ea⊗b). (65)

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Quantum Port-Hamiltonian Network Theory 29

L··· ,k+1 L··· ,k+1

L··· ,k−1 L··· ,k−1

Mj−1,k Mj,k Mj+1,k

√22 dk

dz = dj = 1

time

+

+

gm∆Vb

V +b |1〉

V −b |0〉

gm∆Va

V +a |1〉

V −a |0〉

Fig. 5 RLC circuit equivalent of the Hadamard Gate in the dual-rail encoding [41],

where the solitons propagate in the z-direction. The current across the resistor(s) is

IR(t) = |Ij | sin(ϕj), the current across the capacitor(s) is IC(t) = iωjCj |Vj | cos(ϕj), and

ϕj(t) = ωjt is the RLC quantum phase, and Mj,k is the mutual inductance between the

(k+ 1)-th (upper) RLC quantum channel and the (k−1)-th (lower) RLC quantum channel.

Not drawn to scale (color online).

Definition 4 The composition of Dirac structures Da and Db, denoted Da||Db

is [40]

Da||Db :={

(fI,a, eI,a, fV,a, eV,a, fP,a, eP,a, fI,b, eI,b, fV,b, eV,b, fP,b, eP,b)

∈ Tχ,I,aχI,a ⊗ T †χ,I,aχI,a ⊗ Tχ,V,aχV,a ⊗ T†χ,V,a

χV,a ⊗FP,a ⊗F†P,a

⊗ Tχ,I,bχI,b ⊗ T †χ,I,bχI,b ⊗ Tχ,V,bχV,b ⊗ T†χ,V,b

χV,b ⊗FP,b ⊗F†P,b∣∣∣ ∃ (fa, ea, fb, eb) ∈ Fa⊗b ⊗F†a⊗b

s.t. (fI,a, eI,a, fV,a, eV,a, fP,a, eP,a, fa, ea) ∈ Da

and (−fb, eb, fI,b, eI,b, fV,b, eV,b, fP,b, eP,b) ∈ Db}

(66)

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30 Frederick Ira Moxley III

Mj−1,k Mj,k Mj+1,k

Lj−1,k+1 Lj,k+1 Lj+1,k+1

Lj−1,k−1 Lj,k−1 Lj+1,k−1

√22 dk

√22 dk

√22 dk

√22 dk

+

+

+

+

gm∆Vc

V +c |1〉

V −c |0〉

gm∆Vb

V +b |1〉

V −b |0〉

gm∆VdV +d |1〉

V −d |0〉

gm∆Va

V +a |1〉

V −a |0〉

(a) RLC Circuit CNOT Gate

eP,afP,a

eP,d

fP,d

eP,c

fP,c

χ(Ia)

eI,a

fI,a

fI,beI,b

fV,beV,b

fI,deI,deV,c

fV,c

eI,c

fI,c

eV,d

fV,d

χ(Va)

eV,a

fV,a

eP,bfP,b

fa

fbfc fd

ea

eb

ec ed

χ(Ib)

Db

Dc Dd

Pa

|a〉in

Pb

|b〉in

Pd |d〉inPc|c〉in

χ(Vb)

χ(Id)

χ(Vd)

χ(Vc)

χ(Ic)

Da

(b) CNOT Dirac Structure

Fig. 6 (a) RLC circuit equivalent of the CNOT gate along the z-direction, Eq. (18), where

the current across the resistor(s) is IR(t) = |Ij | sin(ϕj), the current across the capacitor(s) is

IC(t) = iωjCj |Vj | cos(ϕj), and ϕj(t) = ωjt is the RLC quantum phase, and M is the mutual

inductance [31] and (b) Interconnected quantum port-Hamiltonian system representing the

RLC CNOT gate, i.e. the composition of Dirac structures Da, Db, Dc, and Dd. Not drawn

to scale (color online).

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Quantum Port-Hamiltonian Network Theory 31

3.3 Electrostatic Potential Energy

In §2 we considered the case when only one soliton (charge Q dipole) is present.

We now turn our attention to the dipole-dipole electrostatic potential energy

between a point charge dipole at site j and a point charge dipole at site k,

where the mutual inductance in Eq. (49) is given by

Mj,k =1

8πε0

QjQk|rj − rk|

[nj · nk − 3(njk · nj)(njk · nk)], (67)

where ε0 is the is the vacuum permittivity, and

njk =rj − rk|rj − rk|

(68)

and nj is the unit vector pointing along the soliton dipole axis at site j. For

the case of solitons where the inductance dipole-inductance dipole interaction

is the coupling between transition dipoles, Mjk is sometimes referred to as the

exchange energy as it is the energy which characterizes the rate a soliton is

transferred between sites j and k. For the case when the solitons are aligned

parallel to each other, as seen in Figure 1(a), and the unit vector Eq. (68) is

perpendicular to the inductance dipole vector, one obtains

nj · nk = 1, (69)

and

njk · nj = njk · nk = 0. (70)

As such Eq. (67) then becomes

Mj,k =1

8πε0

QjQk|rj − rk|

. (71)

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32 Frederick Ira Moxley III

Furthermore, it should be pointed out that if all of the point charges are the

same then

Qj = Qk = Q, (72)

and Eq. (71) is then

Mj,k =Q2

|rj − rk|. (73)

The charge dipole orientation vectors nj can be described in terms of the

radial unit vector (the direction in which the radial distance from the origin

increases) i.e.,

nj = sin(θj) cos(φj )i + sin(θj) sin(φj )j + cos(θj)k (74)

where θj , and φj are the polar and azimuthal angles, respectively, for the

charge dipole at site j. Similarly, the orientation vector for site nk is given by

nk = sin(θk) cos(φk )i + sin(θk) sin(φk )j + cos(θk)k. (75)

Thus the dot product found in Eq. (67) is written

nj · nk = sin(θj) cos(φj) sin(θk) cos(φk)

+ sin(θj) sin(φj) sin(θk) sin(φk)

+ cos(θj) cos(θk). (76)

The distance vector between two sites rj and rk is given by

rj − rk = (xj − xk )i + (yj − yk )j + (zj − zk)k. (77)

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Quantum Port-Hamiltonian Network Theory 33

The direction vector between sites j and k as seen in Eq. (68) can be written

njk = nxjk i + nyjk j + nzjkk, (78)

where

nxjk =xj − xk√

(xj − xk)2 + (yj − yk)2 + (zj − zk)2, (79a)

nyjk =yj − yk√

(xj − xk)2 + (yj − yk)2 + (zj − zk)2, (79b)

nzjk =zj − zk√

(xj − xk)2 + (yj − yk)2 + (zj − zk)2. (79c)

Using Eqs. (79a)-(79c), one can write

njk · nj = nxjk sin(θj) cos(φj)

+ nyjk sin(θj) sin(φj) + nzjk cos(θj) (80a)

njk · nk = nxjk sin(θk) cos(φk)

+ nyjk sin(θk) sin(φk) + nzjk cos(θk) (80b)

to obtain the value of the product (njk ·nj)(njk ·nk) on the RHS of Eq. (67).

Using Eq. (67) with Eq. (49) yields a Schrodinger equation for the universal

RLC quantum circuit, namely

idαj(t)

dt=

1

2

∑j

[Lj

(Ij−1 − Ij+1

2

)2

+1

Cj

(Qj+1 −Qj−1

2

)2 ]αj(t)

+∑kj 6=k

QjQk|rj − rk|

[nj · nk − 3(njk · nj)(njk · nk)]αk(t).

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34 Frederick Ira Moxley III

4 Conclusion

In this study, we investigated the use of Resistor-Inductor-Capacitor (RLC)

circuits for acheiving universal quantum simulators. These universal quantum

simulators are effectively the 2D Heisenberg and XY models with variable

inductances that can simulate any stoquastic Hamiltonian, where variable in-

ductors simulate the quantum interactions. This was accomplished by devel-

oping the universal gates with RLC circuits, obtained from quantum port-

Hamiltonian network theory. These Dirac structures can then be networked

together such as to execute arbitrary quantum algorithms using the intercon-

nected universal RLC circuit gates. Owing to Digital-to-Analog Conversion

(DAC), our framework admits a fully-programmable architecture, as the pro-

grammable input currents simulate the prepared quantum states for any given

algorithm. The resulting solution is given by the output voltage probabil-

ity amplitudes of the interconnected Dirac structures via Analog-to-Digital

Conversion (ADC). Our construction can perform at high speeds, and stored

(written) electronically, where the logical value 1 (high voltage) or logic 0

(low voltage) is driven into the bit line of a Random-Access Memory (RAM)

memory cell. Due to the stability and high readout or transfer speeds of RLC

circuits, we envision these systems as universal quantum simulators for arbi-

trary quantum algorithms.

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Quantum Port-Hamiltonian Network Theory 35

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