Quantum Port-Hamiltonian Network Theory

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Quantum Port-Hamiltonian Network Theory

Frederick Moxley

To cite this version:

Frederick Moxley. Quantum Port-Hamiltonian Network Theory. rspa.royalsocietypublishing.org Re-

search, In press. �hal-02554914�

rspa.royalsocietypublishing.org

Research

Article submitted to journal

Keywords:

Dirac structures, universal quantum computers, quantum Turing machines

Author for correspondence:

Frederick Ira Moxley III e-mail:

Frederick.I.Moxley@Dartmouth.edu

Quantum Port-Hamiltonian Network Theory

Frederick Ira Moxley III

1Dartmouth College, Hanover, NH 03755 USA

Herein, we propose using Resistor-Inductor-Capacitor (RLC) circuits for achieving universal quantum computers. This is accomplished by explicitly presenting the required set of universal gates using RLC circuits obtained from quantum network theory, where the interaction of the system with its environment is described by an external port. Our proposal for universal quantum information processing systems is based on implementing composite Dirac structures using RLC circuits. These Dirac structures can then be networked together such as to execute arbitrary quantum algorithms using the interconnected universal RLC circuit gates, where variable inductors control the quantum interactions. Owing to Digital-to-Analog Conversion (DAC), our framework admits a fully- programmable architecture, as the programmable input currents are the prepared quantum states for any stoquastic Hamiltonian. The resulting solution of the stoquastic Hamiltonian 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 construction 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 computers to realize the execution of arbitrary quantum stoquastic Hamiltonians.

© The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/

by/4.0/, which permits unrestricted use, provided the original author and source are credited.

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1. Introduction

Following some earlier efforts [1,2], more recently there have been efforts to develop a quantum Turing machine (QTM), or universal quantum computer [3], related to classical and probabilistic Turing machines in a framework based on transition matrices [4]. QTMs 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 arbitrarily different many-body physics models [5]. Herein, we demonstrate that a QTM can be attained by using analog components such as Resistor- Inductor-Capacitor (RLC) circuits [6]. These RLC circuits can be assembled in a fashion which emulates the behavior of a true quantum system [7,8] (e.g. integer factorization [9,10]). This is useful for many applications, including but not limited to, quantum cryptography, and the efficient (i.e., polynomial-runtime) implementation of Universal models of computation. A faithful QTM must obey conservation laws (e.g. energy, charge, etc.) while they exhibit certain dynamical features such as quantum discord [11], and entanglement [12]. In order to perform efficient quantum simulation of fermionic and frustrated systems (while avoiding the exponential growth of statistical errors as the number of particles increases [13]), the QTM must overcome the infamous sign problem [14]. Recently, it has also been rigorously demonstrated in the peer- reviewed academic literature that a fully-programmable QTM can be realized by using the interconnection of universal gates (e.g., coupled qubits [15,16], quantum walks [17,18], two- level systems [19]). Following these and other motivations, herein we present a novel method for constructing an analog QTM using RLC circuits. These circuits scale linearly in the number of classical circuit elements necessary for preparing and controlling an exponential number of quantum states [20], and are essentially quantum Turing machines. This allows for the implementation of universal models of computation, by providing the three elementary gates necessary to do so [21]. 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 QTM. One can also achieve the QTM 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 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 identical 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 functioning as a quantum Turing machine. Moreover, we address the coupling of quantum channels (registers) [22], separated by an arbitrary distance such as to mediate voltage supertransport, where the interaction between quantum channels is mediated by the programmable mutual inductances [23]. In doing so, we successfully control universal quantum walks, thus allowing for programmable quantum computing operations. By considering a quantum network with RLC circuits, an arbitrary universal quantum walk can be programmed. Another important component of any quantum algorithmic implementation is to consider the initial state preparation, and the I/O with respect to that initial state. The RLC analog QTM 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 presenting a novel quantum Port-Hamiltonian theory, which portrays the voltage I/O conversion process as an interconnection of Dirac structures (i.e., rigorously and well- defined mathematical objects). In this case, the Dirac structures represent the analog QTM-RLC gates in terms of effort and flow variables (voltages and currents, respectively).

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The purpose of this article is to investigate the prospects of using RLC circuits to construct universal quantum computers, and to formulate a theoretical description of its operation using time-dependent composite Dirac structures. In §2we introduce the idea of using RLC circuits to prepare quantum analog voltage states, and present analytical solutions for the bright and dark states. We then obtain the quantized energy relation from the electromagnetic radiation of the RLC circuit, and derive the RLC Bell-states from the Shockley diode equation. In §3we present the quantum port-Hamiltonian framework, and illustrate its application using RLC circuits. These RLC circuits are then used to construct the primitive gateset for universal quantum computation, i.e., the phase-shift, Hadamard, and CNOT gates, and are synthesized with the composite Dirac structures of the quantum port-Hamiltonian framework. Finally, a Schrödinger equation for the universal RLC quantum circuit is obtained, and concluding remarks are made in §4.

(a) Preliminaries

Definition 1. LetP1∈C^n×nbe invertible and self-adjoint, letP0∈C^n×nbe skew-adjoint, i.e.,P₀^†=

−P0, and let H ∈L^∞([j= 0, . . . , N];C^n×n), wherej∈N such thatH^†_j=Hj,mI≤ Hj≤M I for a.e.j∈[0, N]and constantsm, M >0independent ofj[24]. We equip the Hilbert spaceX:=L²([j= 0, . . . , N];Cⁿ)with the discrete inner product

hψ, ϕi_X=1

4hϕ0|H0|ψ0i+1 2

N−1

X

j=1

hϕj|Hj|ψji+1

4hϕ_N|H_N|ψ_Ni. (1.1) Then the linear, first-order Schrödinger equation

i~|ψ(t)i˙ =P1

2 X

j

Hj+1− Hj−1

|ψj(t)i

+P1

2 X

j

Hj

|ψj+1(t)i − |ψj−1(t)i

+P0

X

j

Hj|ψj(t)i (1.2)

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 timetis

|ψ(t)i=X

j

αj(t)|ji (1.3)

where the complex numberα_j(t)is

α_j(t) =hj|ψ(t)i, and hj⁰|ji=δ_jj⁰. (1.4) Definition 2. Consider a finite-dimensional linear spaceF ∈C^kwithE=F^†. A subspaceD ⊂ F ⊗ Eis a Dirac Structure if [25]:

(i)he|fi= 0,∀(f, e)∈ D, (ii) dimD=dimF.

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Definition 3. Consider a state space manifoldχand a HamiltonianH :χ→C, defining energy-storage [26], e.g. current I or voltageV. A quantum port-Hamiltonian system onχis defined by the Dirac structure

D ⊂ Tχχ⊗ Tχ^†χ⊗ F_P⊗ E_P, (1.5) having energy-storing port(f_S, e_S)∈ Tχχ⊗ Tχ^†χ, and an external structureP, e.g. source voltage or output current, such that

P ⊂ F_P⊗ E_P, (1.6)

corresponding to an external port(f_P, e_P)∈ F_P⊗ E_P. The temporal dynamics of the quantum system are then specified by

n

˙

αj(t) = ∂

∂πj

hHi,−π˙j(t) = ∂

∂αj

hHi, f_P(t), e_P(t) o

∈ D αj(t)

, t∈R; (1.7) whereα_jare the generalized coordinates, andπ_jare the conjugate momenta.

Remark 1.1. The propertyD=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 Definition2.

Remark 1.2. In the infinite-dimensional case (Ibid.), the propertyD=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 [5]. 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 [14].

2. The RLC Quantum Channel

(a) Quantum State Preparation

We begin by adapting some classical theory of the Operational Transconductance Amplifier (OTA) [27], for purposes of quantum state preparation with universal RLC circuit design, as seen in Figureiii. The transconducting gaingmis proportional to the external dc bias currentIext, where the proportionality constant~is dependent upon temperature, device geometry, and the process [28]. Furthermore, we assume the input and output impedancesZhave ideal values of infinity, i.e.Av=R_in=∞andRout= 0. As such,

gm=~Iext. (2.1)

Controllability of the gain gm, and hence the quantum state preparation can be obtained by programmingIextusing Digital-to-Analog Conversion (DAC) techniques [29]. Furthermore, the output current of an OTA is proportional to the input signal voltage [30], such that

Iout(t) =gm

V_in⁺(t)−V_in⁻(t)

, (2.2)

where the complex-valued voltages are

V_j⁺(t) =|V⁺|exp(iω⁺_jt), (2.3a) V_j⁻(t) =|V⁻|exp(iω_j⁻t); (2.3b)

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(DAC) |ψ

i D(t) = P

j

U (t) |ψ

(0)i (ADC)

e.g. (S ∨ D) -RAM dz = dj = 1

+

−

gm(V⁺|1i −V⁻|0i)

V

|0i V

|1i

L

L

L

Q BL

Q

BL

W L

(a)

RLC Quantum Channel

e

= −f

f

= e

e

= V = ˙ ϕ f

= I = ϕ/L

− he

|f

i = hf

|e

i

he

|f

i + he

|f

i + he

|f

i = 0 7→ H ˙ ≤ he

|f

i

χ _{(I )} D

P

χ _{(V )}

e

= f

f

= −e

(b)

Quantum Port-Hamiltonian System

Figure 1.(a) RLC circuit equivalent of thequantum channel. We work in natural units, whereR=~^{= 1,L=C=t=} 1/eV,ωj= 2πeV, andI= 1eV. Digital-to-Analog Conversion (DAC) is used to prepare the analog quantum state, which then propagates along thez-direction. The time-dependent Dirac structureD(t)then performs a unitary evolution of the prepared quantum state. The current across the resistor(s) isIR(t) =|I_j|sin(ϕj), the current across the capacitor(s) isIC(t) =iωjCj|V_j|cos(ϕj), andϕj(t) =ωjtis the RLC quantum phase. The output of the Dirac structure then undergoes Analog-to-Digital Conversion (ADC), such that the quantum information is stored (cf. measurement) in Random-Access Memory (RAM) cell(s). (b) Quantum Port-Hamiltonian Systemrepresentation of the RLC quantum channel (a), i.e. the port interconnection of currentIand voltageV, the external portP, whereχ(I)is the current storage state,χ(V) is the voltage storage state, the Dirac structureDlinks the storage ports (flows{f_I, fV}and efforts{e_I, eV}, respectively), with the external port (flowfPand efforteP). The electric power-balance (conservation) equation isH˙=i~^∂tt, such that the total power is equal to zero (color online).

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andω^±_jtare the angular frequencies. When an ideal resistorR_L=Z₀is connected to the output of an OTA, a simple voltage amplifier is obtained:

Vout

V_in⁺−V_in⁻ =gmRL. (2.4)

Next we write the complex-valued current as

I=|Ij|exp[i(ω_It−φ)]. (2.5)

With Eqs. (2.3a)-(2.3b) and after DAC, the j-th output current (i.e. prepared analog quantum superposition state) of an OTA in the digital basis is:

I_j^out(t) =gm

|V_j⁺|exp[iω⁺_jt]|1i − |V_j⁻|exp[iω⁻_jt]|0i

, (2.6)

where the logical value(s) 1 represents high voltage(s), and the logical value(s) 0 represents the low voltage(s). By combining Eqs. (2.5)-(2.6), we then obtain thej-th RLC circuit site current with continuous (time-dependent) phases

|Ij^out(t)|= gm

exp[i(ω_It−φ)]

|V_j⁺|exp[iω⁺_jt]|1i − |V_j⁻|exp[iω⁻_j t]|0i

. (2.7)

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)i=|I_j^out(t)i, such that

|ψ(t)i₊= q

I_j⁺exp[i(ω⁺_I t−φ⁺)]|1i, (2.8a)

|ψ(t)i₋= q

I_j⁻exp[i(ω⁻_I t−φ⁻)]|0i; (2.8b) and the superposition state |ψ(t)i=|ψ(t)i₊± |ψ(t)i₋. Hence, from Ohm’s law, the output voltage amplifier has prepared for ADC the OTA output voltage state equation

Vout(t) =RLI_j^out(t). (2.9)

As seen in Figure 1(a), in natural units, where R=~= 1, L=C=t= 1/eV, ωj= 2π eV, and I= 1eV, by takingC= 2/eV we finally obtain the coupled Schrödinger equations governing the electrodynamics of the RLC quantum channel [31–33]

i~V˙j(t)|ψ(t)i₊= 1 2m

h

I_j−1(t)−Ij+1(t)i

|ψ(t)i₊− |Ij|sin2π ϕ0

ϕj(t)

|ψ(t)i₋, i~V˙_j(t)|ψ(t)i₋= 1

2m h

I_j−1(t)−I_j+1(t)i

|ψ(t)i₋− |I_j|sin2π ϕ₀ϕ_j(t)

|ψ(t)i₊; (2.10) whereIj(t)is the current,Vj(t)is the voltage, andϕj(t)is the RLC quantum phase, i.e.

I˙j(t) =− 1 2Lj

[Vj+1(t)−V_j−1(t)], V˙j(t) =− 1

2Cj

h

Ij+1(t)−Ij−1(t) + 2|Ij|sin2π ϕ0

ϕj(t)i ,

˙

ϕ(t) =Vj(t). (2.11)

Furthermore, here it should be pointed out that the position observablexˆ=z=j∆z=j, and the momentum observable pˆ=−i~∂^1/2/∂z^1/2, thereby satisfying the Heisenberg uncertainty principle [34]

hˆx²i hˆp²i ≥~

4. (2.12)

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(i) Bright States

The first solution of Eq. (2.11) is a bright soliton propagation along the RLC quantum channel as seen in Figure1(a). As such, the analytical solution is

ϕj(t) = 2ϕ0

π atanh

exp j−vt λ

q 1−^v_c²2

i (2.13a)

Vj(t) =ϕ0

2πϕ˙j(t)

=−ϕ0ω 2π

2v q

1−^v_c2²

sechh j−vt λ

q 1−^v_c2²

i

(2.13b)

Ij(t) =− ϕ₀ 4πL_j

ϕj+1(t)−ϕ_j−1(t)

=− ϕ₀ 2πL_jλ

2 q

1−^v_c2²

sechh j−vt λ

q 1−^v_c²2

i

(2.13c)

(ii) Dark States

The second solution of Eq. (2.11) is a dark soliton propagation along the RLC quantum channel as seen in Figure1(a). As such, the analytical solution is

ϕ_j(t) = 2ϕ0

π atanh

exp j−vt λ

q 1−^v_c2²

i

(2.14a)

Vj(t) =ϕ₀ 2πϕ˙j(t)

=−ϕ0ω 2π

2v q

1−^v_c²2

tanhh j−vt λ

q 1−^v_c²2

i (2.14b)

Ij(t) =− ϕ0

4πLj

ϕj+1(t)−ϕj−1(t)

=− ϕ0

2πLjλ 2 q

1−^v_c2²

tanh

h j−vt λ

q 1−^v_c²2

i

(2.14c)

(iii) Electromagnetic Radiation of the RLC Quantum Circuit

The quantization of the electromagnetic radiation in the quantum RLC circuit is acheived by consideringπ_j⁰ andqjas 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, πj⁰] = [qj, qj⁰] = 0, [qj, πj⁰] =i~δj,j⁰. (2.15)

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We then define the creation operatorˆa^†_j(t)and the annihilation operatorˆaj(t)by using Eq.

(3.26) such that ˆ a^†_j(t) =

s 1

2~ω_j[ω_jq_j(t)−iπ_j(t)]

= s 1

2~ω_j Z_tf

t0

h ωj

∂

∂π_j hH(αj, πj, t⁰)i+i ∂

∂α_jhH(αj, πj, t⁰)ii dt⁰

= s 1

2~ωj

Zt_f t₀

h∂

∂t

ωjαj(t⁰)−iπj(t⁰)i dt⁰

= s 1

2~ωj

Zt_f t₀

h∂

∂t

ωjαj(t⁰) +~α¯j(t⁰)i dt⁰

= s 1

2~ωj

h

ωjαj(t) +~α¯j(t)i

, (2.16a)

ˆ aj(t) =

s 1 2~ωj

[ωjqj(t) +iπj(t)]

= s

1 2~ωj

Zt_f t0

h ωj

∂

∂πj

hH(αj, πj, t⁰)i −i ∂

∂αj

hH(αj, πj, t⁰)ii dt⁰

= s

1 2~ωj

Zt_f t0

h∂

∂t

ω_jα_j(t⁰) +iπ_j(t⁰)i dt⁰

= s

1 2~ω_j

Ztf

t0

h∂

∂t

ω_jα_j(t⁰)−~α¯_j(t⁰)i dt⁰

= s 1

2~ω_j h

ω_jα_j(t)−~α¯_j(t)i

. (2.16b)

The formal analogy between the operators aˆ^†_j, and ˆa_j 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 eigenfunctionΦ, which is a product of the eigenfunctions of the individual Hamiltonians~ω_j(ˆa^†_jˆa_j+ 1/2)

Φ=un1un2· · ·=

∞

Y

j=1

unj, (2.17)

whereun_j are a complete orthonormal set of basis functions, and ˆ

a^†_junj=p

n_j+ 1un_j+1, ˆa_junj=√

n_jun_j−1, ˆa^†_jaˆ_junj=n_junj. (2.18) The expectation value of the number operatorˆa^†_jˆa_jis then

hΦ|ˆa^†_jˆa_j|Φi=hn_j|ˆa^†_jˆa_j|n_ji=n_j, (2.19) and is equal to the number of quanta n_j in the j-th mode of the quantum channel. More specifically, a quantum channel of lengthLj along the axis of mode volumeV =dkxdkydkz= (2π/Lj)³inkspace with electric and magnetic field vectors pointing in they- andx- directions, respectively, satisfy the Hamiltonian

H=1 2 Z

V

X

j

(µH_j·H_j+εE_j·E_j)dv. (2.20)

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+

−

R^{· · ·,in}_L

|· · ·i_in ^g···m(V_{· · ·,in}⁺ −V_{· · ·,in}⁻ ) V_{· · ·,in}⁺

V_{· · ·,in}⁻

V· · ·,in

+

−

R^d,in_L

|di_in ^gdm(V_d,in⁺ −V_d,in⁻) V_d,in⁺

V_d,in⁻

V_d,in

+

−

R^c,in_L

|ci_in ^gcm(V_c,in⁺ −V_c,in⁻) V_c,in⁺

V_c,in⁻

Vc,in

+

−

R^b,in_L

|bi_in g^bm(V_b,in⁺ −V_b,in⁻) V_b,in⁺

V_b,in⁻

Vb,in

+

−

R^a,in_L

|ai_in g^am(V_a,in⁺ −V_a,in⁻) V_a,in⁺

V_a,in⁻

Va,in

+

−

R^{· · ·}_L^,out

+

−

R^d,out_L

+

−

R^c,out_L

+

−

R^b,out_L

+

−

R^a,out_L

Da||Db||Dc||Dd||D_···

g^···_m(V· · ·⁺,out−V· · ·⁻,out) g^d_m(V_d,out⁺ −V_d,out⁻ ) g^cm(V_c,out⁺ −V_c,out⁻ ) g^bm(V_b,out⁺ −V_b,out⁻ ) g^a_m(V_a,out⁺ −V_a,out⁻ )

|ai_out⊗ |bi_out

|ci_out⊗ |di_out

|· · ·i_out⊗ |· · ·i_out

Figure 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 structureDa||D_b||Dc||D_d||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). For illustration purposes, and not drawn to scale (color online).

As such,

E(x, t) =iX

j

r

~ω_j

V ε[ˆa^†_j(t)−aˆj(t)] sin(j·kj)ˆx

=i

N

X

j=1

r

~ωj

V ε[ˆa^†_j(t)−ˆa_j(t)] sin j· π

N hnj|ˆa^†_jaˆ_j|nji ˆ x

=i~ r 2

V ε

N

X

j=1

¯

αj(t) sin j· π

2~ωjN[ωjα²_j(t)−~²α¯²_j(t)]

ˆ

x, (2.21a)

H(x, t) =X

j

s

~ωj

V µ[ˆa^†_j(t)−ˆaj(t)] cos(j·kj)ˆy

=X

j

s

~ωj

V µ[ˆa^†_j(t)−ˆa_j(t)] cos j· π

N hnj|ˆa^†_jˆa_j|nji ˆ y

=~ r 2

V µ

N

X

j=1

¯

αj(t) cos j· π

2~ωjN[ωjα²_j(t)−~²α¯²_j(t)]

ˆ

y; (2.21b)

where the wavevectork_j=πn_j/L_j. Upon inserting those quantized electric and magnetic fields into the Hamiltonian Eq. (2.20) using Eqs. (2.16a)-(2.16b), we obtain the time-dependent quantized energy relation for theN-quantum channels

H(t) =ˆ ~c n

N

X

j=1

βj

ˆ

a^†_j(t)ˆaj(t) +1 2

, (2.22)

whereβ²_j∼ω²_jε/c²,n=√

ε, the creation operatorsaˆ^†_j(t), and the annihilation operatorsaˆj(t)are given by Eq. (2.16a), and Eq. (2.16b), respectively.

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(b) Analog Bell-State Preparation

In order to describe the Bell-State preparation as seen in Figureiii[35], we invoke theSchockley diode equationfor describing the mixing of the voltage state equations [36], i.e.

I=I_Sh

exp V_D nV_T

−1i

(2.23) whereI 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, andnis 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 form

|ai_out⊗ |bi_out=R^a,out_L |I_m^a,outi+R^b,out_L |I_m^b,outi +1

2 h

(R^a,out_L )²|Im^a,out, Im^a,outi+ 2R^a,out_L R^b,out_L |Im^a,out, Im^b,outi + (R^b,out_L )²|Im^b,out, I_m^b,outii

, (2.24)

and similarly for|ci_out⊗ |di_out. As such, we obtain the Bell states as depicted in Figureiii:

|Φ^±i= 1

√2(|ai_out⊗ |ci_out± |bi_out⊗ |di_out), (2.25a)

|Ψ^±i= 1

√2(|ai_out⊗ |di_out± |bi_out⊗ |ci_out). (2.25b)

3. Quantum Port-Hamiltonian Networks

Port-Hamiltonian systems have been studied extensively for the case of classical RLC-circuits [37,38]. Herein we aim to extend the port-Hamiltonian framework to RLC quantum networks [6], e.g. Figure1(a), for the purpose of developing universal analog quantum computers [39]. The standard way of modeling the system in Figure1(a)is to start with the configuration of the charge (storage state)Q∈χ, and to write down the classical Hamiltonian of the RLC circuit, i.e.

H(Q, ϕ) = 1

2m(L_jI_j)²+1 2kQ²_j

=1 2

X

j

h Lj

∂

∂tQj

2

+ 1 C_j

∂

∂zQj

2i

=1 2

X

j

h Lj

− ∂

∂zIj

2

+ 1 Cj

∂

∂zQj

2i

=1 2

X

j

"

L_j Ij−1−Ij+1

2

!2

+ 1 C_j

Qj+1−Qj−1

2

!2#

(3.1)

sinceQ˙=−∂zI, the energy stored in an inductorT= (LI)²/2m=ϕ²(t)/2m, and the total electric potential energy stored in a capacitor is given byU=kQ²/2, whereC is the capacitance,V is the electric potential difference, and Qis the charge stored in the capacitor, i.e.m=Landk= 1/C. Quantum port-Hamiltonian networks can be regarded as treating the kinetic and potential energies as interconnected subsystems, both of which store energy. Now suppose we have a set of basis states{|ni}that are discrete, and orthonormal, i.e.

hn⁰|ni=δnn⁰. (3.2)

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The instantaneous state of the RLC quantum circuit at timetcan be expanded in terms of these basis states, viz.

|ψ(t)i=X

n

αn(t)|ni, (3.3)

where

αn(t) =hn|ψ(t)i. (3.4)

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

hH(t)i=hψ(t)|H|ψ(t)i

=1 2

X

j

X

nn⁰

¯

α_n0(t)αn(t)

* n⁰

L_j Ij−1−Ij+1

2

!2

+ 1 C_j

Qj+1−Qj−1

2

!2 n

+ . (3.5) With the generalized canonical coordinateq=αn(t) =Q, and the conjugate momentaπn(t) = i~α¯n(t) =ϕ, it can be seen that

∂

∂α¯n⁰

hH(t)i=hψ(t)|H|ψ(t)i

=1 2

X

j

X

n

αn(t)

* n⁰

Lj

I_j−1−I_j+1 2

!2

+ 1 Cj

Q_j+1−Q_j−1 2

!2 n

+

=1 2

X

j

* n⁰

Lj

I_j−1−Ij+1

2

!2

+ 1 Cj

Qj+1−Q_j−1 2

!2

ψ(t) +

. (3.6)

It is well-known that a state vector|ψ(t)ievolves according to the Schrödinger equation:

i~∂

∂t|ψ(t)i=H|ψ(t)i. (3.7)

As such, in addition to using the orthonormality of the basis states, Eq. (3.6) can be written

∂

∂α¯_n0 hH(t)i=i~∂

∂tα_n0. (3.8)

Similarly,

∂

∂αnhH(t)i=−i~∂

∂tα¯n. (3.9)

Hence, we obtain the quantum mechanical equations of motion for the total system as seen in Figure1(a)as

∂

∂t

"

αn

πn

#

=

"

0 1

−1 0

# " _∂

∂α_n

H(αn, πn)

∂

∂π_n

H(αn, πn)

#

. (3.10)

Moreover, theinput-state-output quantum port-Hamiltonian RLC circuitas seen in Figure1(b)with (input) control voltageu:=e_P=V_in, and output currenty:=f_P=I_outis written

∂

∂t

"

αn

πn

#

=

"

0 1

−1 0

# " _∂

∂α_n

H(αn, πn)

∂

∂π_n

H(αn, πn)

# +

"

1 0

# V_in,

I_out=h 1 0i

" _∂

∂α_n

H(αn, πn)

∂

∂π_n

H(αn, πn)

#

; (3.11a)

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where the skew-Hermitian adjointstructure matrixJ, i.e.J^†=−Jis J=

"

0 1

−1 0

#

. (3.12)

This leads to the system of equations for the current:

Current:







Q˙ = ˙αn(t) =−fI, e_I =_∂α^∂

n

H(αn, πn)

; (3.13)

where the flow−fI∈ FIdenotes the current, and the efforte_I∈ EIis the voltage. The reason for the minus sign in front off_I is that we want the productf_Ie_I to be the incoming power with respect to the port interconnection as seen in Figure 1(b). We obtain similar equations for the voltage:

Voltage:







˙

ϕ = ˙πn=−fV, e_V =_∂π^∂

n

H(αn, πn)

. (3.14)

We then couple the current and the voltage subsystems to each other through the interconnection element as detailed in Figure1(b), using the commutation relations

[αn(t), αn⁰(t)] = [πn(t), πn⁰(t)] = 0, [αn(t), πn⁰(t)] =i~δnn⁰; (3.15) viz.,

Interconnection:







−fI =e_V,

fV =eI. (3.16)

Here it should be pointed out that the RLC quantum channel interconnection can be generalized for an arbitrary number of Dirac structures [24].

(a) Mutually Inducting Quantum RLC Circuits

We now consider the case of coupled quantum channels [40,41], 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 system, i.e.

H(Q, ϕ) =1 2

X

j

h Lj

∂

∂tQj

2

+ 1 C_j

∂

∂zQj

2i

±MX

jk

∂

∂tQj

∂

∂tQ_k(1−δ_jk)

=1 2

X

j

"

Lj

I_j−1−I_j+1 2

!2

+ 1 C_j

Q_j+1−Q_j−1 2

!2#

±MX

jk

I_j−1−Ij+1

2

! I_k−1−I_k+1 2

!

(1−δjk) (3.17)

sinceQ˙=−∂zI, and the mutual inductance between the quantum channels propagating in the z-direction is given by the nonlinear coefficient ofM. Now suppose we havetwo setsof basis states{|ni}and{|mi}that are discrete, and orthonormal, i.e. one set of basis states for each RLC quantum channel

hn⁰|ni=δnn⁰ and hm⁰|mi=δmm⁰. (3.18) The instantaneous state of the coupled RLC quantum channelsaandbat timetcan be expanded in terms of a quantum superposition these basis states, viz.

|ψ(t)i=X

n

αn(t)|ni+X

m

βm(t)|mi, (3.19)

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dz = dj = 1

M

M

M

dz = dj = 1

+

− +

−

gm∆Va

V

|0i V

|1i

g_m∆V_b

V

|0i V

|1i

(a)

Mutually Inducting RLC Quantum Channels

e

= V

= ˙ ϕ

e

= V

= ˙ ϕ

e

= −f

e

= −f

f

= e

f

= e

f

= I

= ϕ

/L

f

= I

= ϕ

/L

he

|f

i + he

|f

i + he

|f

i = 0 7→ H ˙

≤ he

|f

i he

|f

i + he

|f

i + he

|f

i = 0 7→ H ˙

≤ he

|f

i

f

f

e

e

χ _(I

a )

χ _(I

b )

D _a D _b

P _a P _b

χ _(V

a )

χ _(V

b )

e_I,a=f_V,a f_I,a=−eV,a

eI,b=fV,b

fI,b=−eV,b

(b)

Composition of Dirac Structures

Figure 3.(a) RLC circuit equivalent of thecoupled quantum channels propagating along thez-direction, Eq. (2.11), 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), andϕj(t) =ωjtis the RLC quantum phase [33] and (b)Interconnected quantum port-Hamiltonian systemrepresenting the RLC coupled quantum channels, i.e. the composition of Dirac structureDaand Dirac structure D_b(color online).

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where

αn(t) =hn|ψa(t)i and βm(t) =hm|ψb(t)i. (3.20) From the expansion of the state in terms of basis states, the expectation value of the Hamiltonian Eq. (3.17) is

hH(t)i=hψ(t)|H|ψ(t)i

=X

nn⁰

hn⁰|¯αn⁰(t)αn(t)H|ni+X

mn⁰

hn⁰|¯αn⁰(t)βm(t)H|mi

+X

m⁰n

hm⁰|β¯_m⁰(t)αn(t)H|ni+ X

m⁰m

hm⁰|β¯_m⁰(t)βm(t)H|mi. (3.21) With the generalized canonical coordinatesqa=αn(t) =Qa,qb=βm(t) =Qb, and the conjugate momentaπn^a(t) =i~α¯n(t) =ϕa,π^bm(t) =i~β¯m(t) =ϕb, it can be seen that

∂

∂α¯_n0 hH(t)i=hn⁰|Hαn(t)|ni+hn⁰|Hβm(t)|mi

=hn⁰|H|ψa(t)i+hn⁰|H|ψb(t)i, (3.22a)

∂

∂β¯m⁰

hH(t)i=hm⁰|Hαn(t)|ni+hm⁰|Hβm(t)|mi

=hm⁰|H|ψa(t)i+hm⁰|H|ψ_b(t)i; (3.22b)

∂

∂αn hH(t)i=hn⁰|¯α_n0(t)H|ni+hm⁰|β¯_m0(t)H|ni

=hψa(t)|H|ni+hψb(t)|H|ni, (3.22c)

∂

∂βmhH(t)i=hn⁰|¯αn⁰(t)H|mi+hm⁰|β¯m⁰(t)H|mi

=hψa(t)|H|mi+hψ_b(t)|H|mi. (3.22d) For the case of coupled RLC quantum channels, the state vector|ψ(t)ievolves according to the coupled nonlinear Schrödinger equations:

i~∂

∂t|ψa(t)i=1 2

X

j

"

L_j I_j−1−Ij+1

2

!2

+ 1 Cj

Qj+1−Q_j−1 2

!2#

|ψa(t)i

±MX

jk

I_j−1−I_j+1 2

! I_k−1−Ik+1

2

!

(1−δ_jk)|ψ_b(t)i, (3.23a)

i~∂

∂t|ψ_b(t)i=1 2

X

j

"

Lj

I_j−1−I_j+1 2

!2

+ 1 Cj

Q_j+1−Q_j−1 2

!2#

|ψ_b(t)i

±MX

jk

I_j−1−Ij+1

2

! I_k−1−I_k+1 2

!

(1−δjk)|ψa(t)i; (3.23b) where the+M corresponds to abrightsoliton solution, and the−Mcorresponds to adarksoliton solution. As such, in addition to using the orthonormality of the basis states, Eqs. (3.22a)-(3.22d) can be rewritten

∂

∂α¯n⁰

hH(t)i=i~∂

∂tαn⁰, ∂

∂β¯_m0

hH(t)i=i~∂

∂tβm⁰. (3.24)

Similarly,

∂

∂αn

hH(t)i=−i~∂

∂tα¯n, ∂

∂βm

hH(t)i=−i~∂

∂tβ¯m. (3.25)

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

∂

∂t









 αn

πn^a

βm

π^bm











=











0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0



























∂

∂α_n

D

H(αn, π^a_n, βm, π^b_m)E

∂

∂π_n^b

D

H(αn, π_n^a, βm, π^b_m)E

∂

∂β_m

D

H(αn, πn^a, βm, π^bm) E

∂

∂π^b_m

D

H(αn, πn^a, βm, π^bm)E

















. (3.26)

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

Currenta :







Q˙a = ˙αn(t) =−fI,a, e_I,a =_∂α^∂

n

D

H(αn, π^a_n, βm, π_m^b )E

, (3.27a)

Currentb :







Q˙_b = ˙βm(t) =−f_I,b, eI,b =_∂β^∂

m

D

H(αn, π^an, βm, πm^b )E

; (3.27b)

where the flows −f_I,a∈ Fa, and−f_I,b∈ F_bdenote the currents, and the effortse_I,a∈ Ea, and e_I,b∈ E_b, are the voltages. The reason for the minus signs in front of thef_I’s is that we want the productsfI,aeI,aandf_I,be_I,bto be the incoming powers with respect to the port interconnections.

We then obtain similar equations for the voltages:

Voltage_a :







˙

ϕa= ˙π^a_n=−fV,a, e_V,a=_∂π^∂a

n

D

H(αn, π_n^a, βm, π^b_m)E

, (3.28a)

Voltage_b :







˙

ϕ_b= ˙πm^b =−f_V,b, eV,b=_∂π^∂_b

m

D

H(αn, πn^a, βm, π^bm)E

. (3.28b)

We then couple the current and the voltage subsystems to each other through the interconnection element using the commutation relations

[αn(t), αn⁰(t)] = [π^an(t), π^a_n0(t)] = 0, [αn(t), π^a_n0(t)] =i~δnn⁰, (3.29a) [βm(t), β_m⁰(t)] = [π^bm(t), π_m^b0(t)] = 0, [βm(t), π_m^b0(t)] =i~δ_mm⁰; (3.29b) viz.,

Interconnection:















−f_I,a =e_V,a, f_V,a =e_I,a;

−f_I,b =e_V,b, f_V,b =e_I,b.

(3.30)

(b) Composite Dirac Structures

We now consider a Dirac structureDaon a product spaceFa⊗ F_a⊗bof two linear spacesFaand F_a⊗b, and another Dirac structureDbon a product spaceF_a⊗b⊗ Fb, whereFbis also a linear space. The linear spaceFa⊗b is the space of shared flow variables, i.e.{fa, f_b}, andF_a⊗b^† is the the space of shared effort variables, i.e.{ea, e_b}as shown in Figure3(b). In order to interconnect DawithD_b, the sign convention for the power flow corresponding to(f_a⊗b, e_a⊗b)∈ F_a⊗b⊗ F_a⊗b^† must first be addressed.

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V

|0i V

|1i δ

∆

~ ω ~ ω(1 − ∆)

~ ω(1 + δ)

(a)

Energy-Level Diagram

dz = dj = 1

L_j−1 Mj L_j+1

~ ω(1 + δ − ∆)

dz = dj = 1

+

−

+

−

gm∆Va

V_a⁻|0i V_a⁺|1i

gm∆Vb

V_b⁻|0i V_b⁺|1i

(b)

RLC Quantum Phase-Shift Gate

Figure 4.RLC circuit equivalent of thePhase-Shift Gatein the dual-rail encoding [43], where the soliton propagates in thez-direction, as described by Eq. (2.11). The current across the resistor(s) isIR(t) =|I_j|sin(ϕj), the current across the capacitor(s) isIC(t) =iωjCj|V_j|cos(ϕ_j), andϕj(t) =ωjtis the RLC quantum phase, andM is the mutual inductance. The phase-shift is acheived by tuning the inductancesδLand∆L, such that|bi_out= exp(iδ−i∆)|bi_in. Not drawn to scale (color online).

Takinghe|fias the incoming power, then since

(fI,a, eI,a, fV,a, eV,a, f_P,a, e_P,a, fa, ea)∈ Da

⊂ Tχ,I,aχI,a⊗ T_χ,I,a^† χI,a⊗ Tχ,V,aχV,a⊗ T_χ,V,a^† χV,a

⊗ F_P,a⊗ F_P,a^† ⊗ F_a⊗b⊗ F_a⊗b^† (3.31)

we have the power coming intoDa denoted ashea|fai owing to the flow and effort variables (fa, ea)∈ F_a⊗b⊗ F_a⊗b^† . Similarly,

(f_b, e_b, f_I,b, e_I,b, f_V,b, e_V,b, f_P,b, e_P,b)∈ D_b

⊂ F_a⊗b⊗ F_a⊗b^† ⊗ T_χ,I,bχ_I,b⊗ T_χ,I,b^† χ_I,b

⊗ T_χ,V,bχ_V,b⊗ T_χ,V,b^† χ_V,b⊗ F_P,b⊗ F_P,b^† , (3.32)