Binary Full-Adder in a Single Quantum System

Abstract

In this short book chapter we show how to implement a complex Boolean function in a simple quantum system using the quantum Hamiltonian computing approach (QHC) [Ref: J. Phys. A: Math. Theor. 44 (2011) 155302 (15pp)]. Following the QHC approach, the logical inputs are encoded in local modifications of the system’s Hamiltonian, and the outputs are read in the oscillation frequency of the population of well-defined target states. Few simple examples are presented first introducing a graphical aid that facilitates the design of QHC circuits. Using this graphical analogue to our symbolic analysis, we demonstrate how to easily implement multiple-inputs multiple-outputs logic gates taking benefit from the superposition principle. A binary full-adder is presented using this generalization of the QHC approach. We also show that using their nonlocal effect, each logical input needs to appear only once in the system and that different logical outputs are computed simultaneously.

Appendices

A.1 Simple Symbolic Analysis of the QHC Approach

A brief overview of the recently developed symbolic analysis of simple quantum systems is presented in this appendix. The development of the general symbolic analysis developed in Sect. 2 is based on this previous work, and the interested reader should refer to the reference [16] for further details. The general form of the Hamiltonian implementing a 1-output logic gate reads:

(A.2)

where $\vert {\phi }_{a}\rangle$ is the initial state of the evolution and $\vert {\phi }_{b}\rangle$ the target state where the logical output status is monitored. If the population of $\vert {\phi }_{b}\rangle$ oscillates with a high frequency, the logical output is 1 and is 0 if this oscillation frequency is low. The Lowdin partitioning of the system is used to access the value of this oscillation frequency, noted Ω in the following. Introducing the projector P on the $\{\vert {\phi }_{a}\rangle ,\vert {\phi }_{b}\rangle \}$ subspace and the projector Q on the rest of the system, the effective Lowdin partitioning reads [22]:

$${\mathcal{H}}_{\mathrm{eff}}({E},\boldsymbol{\alpha}) = \boldsymbol{P}\mathcal{H}(\alpha )\boldsymbol{P} {+\lim }_{\eta \rightarrow 0}P\mathcal{V}Q \frac{1} {E \pm \mathrm{i}\eta -{\mathcal{H}}_{0}(\boldsymbol{\alpha})}Q\mathcal{V}\boldsymbol{P} $$

(A.2)

The evolution generated by this 2 ×2 Hamiltonian smooth the complex evolution going through the ${\mathcal{H}}_{0}(\alpha )$ subspace by a Rabi-like evolution whose oscillation frequency is precisely Ω. An expression of Ω is given by the difference of the two eigenvalues of ${\mathcal{H}}_{\mathrm{eff}}(E,\alpha )$. This expression can be written in term of a Cauchy principal part and a Dirac distribution as:

$$\boldsymbol{\varOmega} (\boldsymbol{\alpha}) = \mathcal{P}\left ({\mathcal{F}}^{-1}({E},\boldsymbol{\alpha} )\right ) \mp \mathrm{i}\pi \delta \big{(}\mathcal{F}({E},\boldsymbol{\alpha})\big{)} $$

(A.3)

where the function $\mathcal{F}({E},\boldsymbol{\alpha})$ comes from the diagonalization of ${\mathcal{H}}_{\mathrm{eff}}({E},\boldsymbol{\alpha})$ and reads:

$$\mathcal{F}({E},\boldsymbol{\alpha}) = \mathcal{A}\times \mathrm{det}\big{(}{E} -{\mathcal{H}}_{0}(\boldsymbol{\alpha})\big{)} $$

(A.4)

with $\mathcal{A}\,=\,\big{[}\sqrt{{({h}_{aa } - {h}_{bb } )}^{2 } + 4{h}_{ab }^{2}}{\big{]}}^{-1}$ and ${h}_{ij}\,=\,\langle {\phi }_{i}\vert V adj[E -{\mathcal{H}}_{0}(\boldsymbol{\alpha})]V \vert {\phi }_{j}\rangle$. The principal part refers to the slow oscillation frequency obtained when $\mathcal{F}({E},\boldsymbol{\alpha})\neq 0$. At the contrary, when $\mathcal{F}({E},\boldsymbol{\alpha})\,=\,0$, the Dirac distribution dominates (1), and this leads to a high oscillation frequency. We are particularly interested by locating the points where $\mathcal{F}({E},\boldsymbol{\alpha})\,=\,0$ since they correspond to the rare values of the phase space where a 1 logical output is obtained. Consequently, we approximate the expression of Ω only considering the Dirac distributions. Our symbolic analysis relies therefore on the expression:

$$\mathfrak{B}({E},\boldsymbol{\alpha}) = \delta \big{(}\mathcal{F}({E},\boldsymbol{\alpha})\big{)} $$

(A.5)

This last expression is however not a symbolic analysis since it does no involve Boolean operators. A decomposition of $\mathfrak{B}({E},\boldsymbol{\alpha})$ over the possible values of the logical inputs can be obtained using the properties of the Fourier transform of Dirac distributions. Using this decomposition, (A.5) is reexpressed as a sum of Dirac distribution weighted by Boolean operators. In its most general form, these operators are symmetric Boolean operators, ${\mathcal{S}}_{i}(\boldsymbol{\alpha})$, that equals 1 if i logical inputs equal one. This expression reads:

$$\mathfrak{B}({E},\boldsymbol{\alpha}) =\sum\limits_{i=0}^{N}{\mathcal{S}}_{ i}(\boldsymbol{\alpha}) \cdot \delta \big{(}\mathcal{F}({E},{\boldsymbol{\alpha}}_{{2}^{i}-1})\big{)} $$

(A.6)

Each Dirac distribution is associated with one of the Boolean operator. Therefore, one or several ${\mathcal{S}}_{i}(\boldsymbol{\alpha})$ operators can be selected by canceling out the argument of their respective distribution.

A.2 Pseudo-Boolean Symbolic Expression in the General Case

The demonstration of (6) can be obtained from the same Lowdin partitioning used in the A.1. However, a simpler and more insightful demonstration is possible studying the properties of the population of the target states. To demonstrate (6), the complete expression of ${\mathcal{P}}_{n}(t)$, when the $\vert {\boldsymbol{\Phi} }_{a}\rangle$ and $\vert {\boldsymbol{\Phi} }_{{b}_{n}}\rangle$ are given by (4) and (5), reads:

$${\mathcal{P}}_{a{b}_{n}}(t) = \frac{1} {L{L}_{{b}_{n}}}{\left \vert \sum\limits_{{\lambda }_{a}=1}^{L}\sum\limits_{\lambda =1}^{{L}_{{b}_{n}} }\langle {\phi }_{b}^{({s}_{n}^{\lambda }) }\vert {\mathrm{e}}^{-\mathrm{i}\mathcal{H}(\alpha )t}\vert {\phi }_{ a}^{({\lambda }_{a})}\rangle \right \vert }^{2} $$

(A.7)

A very fast oscillation is only obtained between two states $\vert {\phi }_{b}^{({s}_{n}^{\lambda }) }\rangle$ and $\vert {\phi }_{a}^{({\lambda }_{a})}\rangle$ with the same energy. Therefore, to have a resonant oscillation, we must have ${s}_{n}^{\lambda }\,=\,{\lambda }_{a}$. Neglecting the low-frequency components of (A.7) which does not respect this last condition leads to:

$${\mathcal{P}}_{a{b}_{n}}(t) \simeq \frac{1} {L{L}_{{b}_{n}}}{\left \vert \sum\limits_{\lambda =1}^{{L}_{{b}_{n}} }\langle {\phi }_{b}^{({s}_{n}^{\lambda }) }\vert {\mathrm{e}}^{-\mathrm{i}\mathcal{H}(\alpha )t}\vert {\phi }_{ a}^{({s}_{n}^{\lambda }) }\rangle \right \vert }^{2} $$

(A.8)

Considering only the cases where only one eigenenergy of ${\mathcal{H}}_{0}(\boldsymbol{\alpha})$ is possibly equal to one ${E}_{l}\,=\,\langle {\phi }_{a}^{(l)}\vert \mathcal{H}\vert {\phi }_{a}^{(l)}\rangle \,=\,\langle {\phi }_{b}^{(l)}\vert \mathcal{H}\vert {\phi }_{b}^{(l)}\rangle$ energy leads to:

$${\mathcal{P}}_{a{b}_{n}}(t) \simeq \frac{1} {L{L}_{{b}_{n}}}\sum\limits_{\lambda =1}^{{L}_{{b}_{n}} }{\left \vert \langle {\phi }_{b}^{({s}_{n}^{\lambda }) }\vert {\mathrm{e}}^{-\mathrm{i}\mathcal{H}(\alpha )t}\vert {\phi }_{ a}^{({s}_{n}^{\lambda }) }\rangle \right \vert }^{2} $$

(A.9)

since the crossed terms $\langle {\phi }_{b}^{(k)}\vert {\mathrm{e}}^{-\mathrm{i}\mathcal{H}(\alpha )t}\vert {\phi }_{a}^{(k)}\rangle \langle {\phi }_{b}^{(l)}\vert {\mathrm{e}}^{-\mathrm{i}\mathcal{H}(\alpha )t}\vert {\phi }_{a}^{(l)}\rangle$ are null unless k = l. We clearly see here that if only one term of this sum is resonant, then the maximum amplitude for small times of (A.9) is ${(L{L}_{{b}_{n}})}^{-1}$. Now, if κ ${\mathcal{H}}_{0}(\boldsymbol{\alpha})$ eigenenergies are equal to κ different E _λ, then very high-frequency components appear in ${\mathcal{P}}_{a{b}_{n}}(t)$ and the maximum amplitude this function reaches for small times is ${\kappa }^{2}/(L{L}_{{b}_{n}})$. For example, if one ${\mathcal{H}}_{0}(\boldsymbol{\alpha})$ eigenenergy is equal to E _i and another one to E _j , then a $1/\hslash \vert {E}_{i} - {E}_{j}\vert$ frequency appears in the ${\mathcal{P}}_{a{b}_{n}}(t)$ spectrum and the small time amplitude of this function increases to $4/(L{L}_{{b}_{n}})$. Such a situation is however not explored in the following.

Since (A.9) is the superposition of elementary terms, the secular oscillation rate, Ω _n , is also the superposition of the secular oscillation rates of each $\big{\vert }\langle {\phi }_{b}^{({s}_{n}^{\lambda }) }\vert {\mathrm{e}}^{-\mathrm{i}\mathcal{H}(\alpha )t}\vert {\phi }_{a}^{({s}_{n}^{\lambda }) }\rangle {\big{\vert }}^{2}$ function given by:

$${\boldsymbol{\varOmega}}_{n}(\boldsymbol{\alpha}) =\sum\limits_{\lambda =1}^{{L}_{{b}_{n}} }\mathcal{P}\left ({\mathcal{F}}^{-1}({E}_{{ s}_{n}^{\lambda }},\boldsymbol{\alpha})\right ) \mp \mathrm{i}\pi \delta \big{(}\mathcal{F}({E}_{{s}_{n}^{\lambda }},\boldsymbol{\alpha})\big{)} $$

(A.10)

Neglecting the principal part in (A.10) to define the pseudo-Boolean symbolic expression of ${\boldsymbol{\varOmega}}_{n}(\boldsymbol{\alpha})$ leads to:

$$\mathfrak{B}(\boldsymbol{\alpha}) =\sum\limits_{\lambda =1}^{{L}_{{b}_{n}} }\delta \big{(}\mathcal{F}({E}_{{s}_{n}^{\lambda }},\boldsymbol{\alpha})\big{)} $$

(A.11)

Using the decomposition of each term of this sum over the symmetric Boolean operators introduced in the previous section leads finally to :

$$\mathfrak{B}(\boldsymbol{\alpha}) =\sum\limits_{i=1}^{M}{\mathcal{S}}_{ i}(\boldsymbol{\alpha}) \cdot \left [\sum\limits_{\lambda =1}^{{L}_{{b}_{n}} }\delta \big{(}\mathcal{F}({E}_{{s}_{n}^{\lambda }},{\boldsymbol{\alpha}}_{{2}^{i}-1})\big{)}\right ] $$

(A.12)

$$\mathcal{F}({E}_{{s}_{n}},\boldsymbol{\alpha}) = \frac{\mathrm{det}\left ({E}_{{s}_{n}} -{\mathcal{H}}_{0}(\boldsymbol{\alpha})\right )} {\sqrt{{({h}_{aa } - {h}_{bb } )}^{2 } + 4{h}_{ab }^{2}}} $$

(A.13)

with ${h}_{ij} =\langle {\phi }_{i}^{({s}_{n})}\vert V \;\mathrm{adj}[{E}_{{s}_{ n}} -{\mathcal{H}}_{0}(\boldsymbol{\alpha})]V \vert {\phi }_{j}^{({s}_{n})}\rangle$. This completes the demonstration of our generalization of the QHC approach to large systems.