n-qubit operations on sphere and queueing scaling limits for programmable quantum computer

Abstract

We derive a general spherical coordinate formula for a quantum state of n-qubit register and study n-qubit operation rules on $(n+1)$-sphere with the target to help developing a (photon or other technique)-based programmable quantum computer. The newly developed angle-based n-qubit operation rules are simple and efficient, which reduce the complicated quantum multiplication and division operations to simple addition and subtraction operations just like those used in a conventional computer. The rules for n-qubit operations are realized through measurement-based feedback controls and quantum entanglements. In the meanwhile, we derive the scaling limits (called reflecting Gaussian random fields on a $(n+1)$-sphere) for n-qubit quantum computer-based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the $(n+1)$-sphere as arrival inputs. In the first regime, the qubit number n is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit in terms of balancing the arrival and service rates under first-in first-out and work-conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number n itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. This regime is in contrast to the well-known Halfin–Whitt regime.

Appendices

Appendix: Proof of Proposition 2.1

The proof of Proposition 2.1 consists of the following two parts that correspond to the two counterparts in the statement of the theorem.

1.1 Proof of Part I

In this part of proof, we prove the claim that a n-qubit quantum wave function $|\Psi \rangle$ satisfies the constraint in (2.2) if its coefficients are given by (2.4). In fact, for the example shown in Fig. 2 with $n=1$, the single-qubit quantum wave function $|\Psi \rangle =\psi _{0}|0\rangle +\psi _{1}|1\rangle$ with $\psi _{0}=\text{ cos }\left( \theta _{1}\right)$ and $\psi _{1}=e^{i\varphi }\text{ sin }(\theta _{1})$ and satisfies $|\psi _{0}|^{2}+|\psi _{1}|^{2}=1$. Furthermore, if $n=2$, the coefficients of the corresponding 2-qubit quantum wave function in (2.1), which is given by

$$\begin{aligned} |\Psi \rangle =\psi _{0}|00\rangle +\psi _{1}|01\rangle +\psi _{2}|10\rangle +\psi _{3}|11\rangle , \end{aligned}$$

also satisfy the relationship in (2.2) due to the following computation,

$$\begin{aligned}&\left| \psi _{0}\right| ^{2}+\left| \psi _{1}\right| ^{2}+\left| \psi _{2}\right| ^{2}+\left| \psi _{3}\right| ^{2} \\&\quad =\left| \cos (\theta _{1})\right| ^{2}+\left| \sin (\theta _{1})\cos (\theta _{2})\right| ^{2} +\left| \sin (\theta _{1})\sin (\theta _{2})\cos (\theta _{3})\right| ^{2}\\&\qquad +\left| e^{i\varphi }\sin (\theta _{1})\sin (\theta _{2})\sin (\theta _{3})\right| ^{2} \\&\quad =\left| \cos (\theta _{1})\right| ^{2}+\left| \sin (\theta _{1})\right| ^{2}\left| \cos (\theta _{2})\right| ^{2}\\&\qquad +\left| \sin (\theta _{1})\right| ^{2}\left| \sin (\theta _{2})\right| ^{2}\left( |\cos (\theta _{3})|^{2}+|\sin (\theta _{3})|^{2}\right) \\&\quad =\left| \cos (\theta _{1})\right| ^{2}+\left| \sin (\theta _{1})\right| ^{2}\left( |\cos (\theta _{2})|^{2}+|\sin (\theta _{2})|^{2}\right) =1. \end{aligned}$$

By the similar way, we can show that the constraint in (2.2) is true for the given expressions in (2.4) corresponding to each $n\in \{1,2,...\}$.

1.2 Proof of Part II

In this part of proof, we prove the claims concerning the four n-qubit quantum operations as stated in part II of Proposition 2.1 to be true. More precisely, we prove the claims by constructing four explicit mapping functions required by (2.7)–(2.8) through explicitly deriving their associated coefficients corresponding to the four operations of addition (+), subtraction (-), multiplication ($*$), and division ($/$) over $S^{n+1}$. For convenience, we summarize them into four steps as follows.

Step I. For the addition (+) operation, the corresponding coefficients of $|\Upsilon \rangle (\Phi ,\Psi ) =|\Upsilon \rangle ^{\Phi +\Psi }$ in (2.7) can be calculated as follows. For $k=1$, we have that

$$\begin{aligned} \upsilon _{1}(\phi _{1},\psi _{1})=\frac{\phi _{1}+\psi _{1}}{2\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) } =\frac{\cos \left( \theta ^{\Phi }_{1}\right) +\cos \left( \theta ^{\Psi }_{1}\right) }{2\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) } =\cos \left( \frac{\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}}{2}\right) .\nonumber \\ \end{aligned}$$

(3.39)

For $k=2$, we have that

$$\begin{aligned} \upsilon _{2}(\phi _{2},\psi _{2})= & {} \frac{(\phi _{2}+\psi _{2})+\sin (\theta ^{\Phi }_{1})\cos (\theta ^{\Psi }_{2}) +\sin (\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2})}{2^{2}\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}}{2}\right) } \nonumber \\= & {} \frac{\left( \sin (\theta ^{\Phi }_{1})+\sin (\theta ^{\Psi }_{1})\right) \left( \cos (\theta ^{\Phi }_{2})+\cos (\theta ^{\Psi }_{2})\right) }{2^{2}\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}}{2}\right) } \nonumber \\= & {} \sin \left( \frac{\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}}{2}\right) . \end{aligned}$$

(3.40)

In general, for an integer $k\in \{2,3,...,2^{n}-1\}$, we have that

$$\begin{aligned} \upsilon _{k}(\phi _{k},\psi _{k})= & {} \frac{1}{2^{k}\prod _{j=1}^{k}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) } \nonumber \\&\times \bigg ((\phi _{k}+\psi _{k})+\Big (\prod _{j=1}^{k-1}\left( \sin (\theta ^{\Phi }_{j})+\sin (\theta ^{\Psi }_{j})\right) \left( \cos (\theta ^{\Phi }_{k})+\cos (\theta ^{\Psi }_{k})\right) \Big ) \nonumber \\&-\Big (\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j})\cos (\theta ^{\Phi }_{k})+\prod _{j=1}^{k-1}\sin (\theta ^{\Psi }_{j})\cos (\theta ^{\Psi }_{k})\Big )\bigg ) \nonumber \\= & {} \frac{\prod _{j=1}^{k-1}\left( \sin (\theta ^{\Phi }_{j})+\sin (\theta ^{\Psi }_{j})\right) \left( \cos (\theta ^{\Phi }_{k})+\cos (\theta ^{\Psi }_{k})\right) }{2^{k}\prod _{j=1}^{k}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) } \nonumber \\= & {} \prod _{j=1}^{k-1}\sin \left( \frac{\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k}}{2}\right) . \end{aligned}$$

(3.41)

Finally, for $k=2^{n}$, we have that

$$\begin{aligned} \upsilon _{2^{n}}(\phi _{2^{n}},\psi _{2^{n}})= & {} \frac{1}{2^{2^{n-1}}e^{\left( i\left( \theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi }\right) /2\right) } \prod _{j=1}^{2^{n}-1}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) } \nonumber \\&\bigg ((\phi _{2^{n}}+\psi _{2^{n}})+e^{i\left( \theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi }\right) } \prod _{j=1}^{2^{n}-1}\left( \sin (\theta ^{\Phi }_{j})+\sin (\theta ^{\Psi }_{j})\right) \nonumber \\&-\Big (e^{i\theta _{2^{n}}^{\Phi }}\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j}) +e^{i\theta _{2^{n}}^{\Psi }}\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Psi }_{j})\Big )\bigg ) \nonumber \\= & {} \frac{e^{i\left( \theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi }\right) } \prod _{j=1}^{2^{n}-1}\left( \sin (\theta ^{\Phi }_{j})+\sin (\theta ^{\Psi }_{j})\right) }{2^{2^{n-1}}e^{\left( i\left( \theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi }\right) /2\right) } \prod _{j=1}^{2^{n}-1}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) } \nonumber \\= & {} e^{\left( i\left( \theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi }\right) /2\right) } \prod _{j=1}^{2^{n}-1}\sin \left( \frac{\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j}}{2}\right) . \end{aligned}$$

(3.42)

Thus, $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi +\Psi }$ in (2.7) with the coefficients in (3.39)–(3.42) satisfies the constraint in (2.2) and has the spherical coordinate $\theta ^{|\Upsilon \rangle ^{\Phi +\Psi }}$ as in (2.8).

Step II. For the subtraction (-) operation, the corresponding coefficients of $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi -\Psi }$ in (2.7) can be calculated as follows. For $k=1$, we have that

$$\begin{aligned} \upsilon _{1}(\phi _{1},\psi _{1})=\frac{\phi _{1}-\psi _{1}+2\cos \left( \theta ^{\Psi }_{1}\right) }{2\cos \left( \frac{\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}}{2}\right) } =\frac{\cos \left( \theta ^{\Phi }_{1}\right) +\cos \left( \theta ^{\Psi }_{1}\right) }{2\cos \left( \frac{\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}}{2}\right) } =\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) .\nonumber \\ \end{aligned}$$

(3.43)

For $k=2$, we have that

$$\begin{aligned} \upsilon _{2}(\phi _{2},\psi _{2})= & {} \frac{(\phi _{2}-\psi _{2})+\sin (\theta ^{\Phi }_{1})\cos (\theta ^{\Psi }_{2}) -\sin (\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2})}{2^{2}\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2}}{2}\right) } \nonumber \\= & {} \frac{\left( \sin (\theta ^{\Phi }_{1})-\sin (\theta ^{\Psi }_{1})\right) \left( \cos (\theta ^{\Phi }_{2})+\cos (\theta ^{\Psi }_{2})\right) }{2^{2}\cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2}}{2}\right) } \nonumber \\= & {} \sin \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}}{2}\right) . \end{aligned}$$

(3.44)

In general, for an integer $k\in \{2,3,...,2^{n}-1\}$, we have that

$$\begin{aligned} \upsilon _{k}(\phi _{k},\psi _{k})= & {} \frac{1}{2^{k}\prod _{j=1}^{k-1}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k}}{2}\right) }\nonumber \\&\times \bigg ((\phi _{k}-\psi _{k})+\Big (\prod _{j=1}^{k-1}\left( \sin (\theta ^{\Phi }_{j})-\sin (\theta ^{\Psi }_{j})\right) \left( \cos (\theta ^{\Phi }_{k})+\cos (\theta ^{\Psi }_{k})\right) \Big ) \nonumber \\&-\Big (\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j})\cos (\theta ^{\Phi }_{k})-\prod _{j=1}^{k-1}\sin (\theta ^{\Psi }_{j})\cos (\theta ^{\Psi }_{k})\Big )\bigg ) \nonumber \\= & {} \frac{\prod _{j=1}^{k-1}\left( \sin (\theta ^{\Phi }_{j})-\sin (\theta ^{\Psi }_{j})\right) \left( \cos (\theta ^{\Phi }_{k})+\cos (\theta ^{\Psi }_{k})\right) }{2^{k}\prod _{j=1}^{k-1}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k}}{2}\right) } \nonumber \\= & {} \prod _{j=1}^{k-1}\sin \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) \cos \left( \frac{\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k}}{2}\right) . \end{aligned}$$

(3.45)

Finally, for $k=2^{n}$, we have that

$$\begin{aligned} \upsilon _{2^{n}}(\phi _{2^{n}},\psi _{2^{n}})= & {} \frac{1}{2^{2^{n-1}}e^{\left( i\left( \theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi }\right) /2\right) } \prod _{j=1}^{2^{n}-1}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) } \nonumber \\&\times \bigg ((\phi _{2^{n}}+\psi _{2^{n}})+e^{i\left( \theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi }\right) } \prod _{j=1}^{2^{n}-1}\left( \sin (\theta ^{\Phi }_{j})-\sin (\theta ^{\Psi }_{j})\right) \nonumber \\&-\Big (e^{i\theta _{2^{n}}^{\Phi }}\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j}) -e^{i\theta _{2^{n}}^{\Psi }}\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Psi }_{j})\Big )\bigg ) \nonumber \\= & {} \frac{e^{i\left( \theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi }\right) } \prod _{j=1}^{2^{n}-1}\left( \sin (\theta ^{\Phi }_{j})-\sin (\theta ^{\Psi }_{j})\right) }{2^{2^{n-1}}e^{\left( i\left( \theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi }\right) /2\right) } \prod _{j=1}^{2^{n}-1}\cos \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) } \nonumber \\= & {} e^{\left( i\left( \theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi }\right) /2\right) } \prod _{j=1}^{2^{n}-1}\sin \left( \frac{\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}}{2}\right) . \end{aligned}$$

(3.46)

Thus, $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi -\Psi }$ in (2.7) with the coefficients in (3.43)–(3.46) satisfies the constraint in (2.2) and has the spherical coordinate $\theta ^{|\Upsilon \rangle ^{\Phi -\Psi }}$ as in (2.8).

Step III. For the multiplication ($*$) operation, the corresponding coefficients of $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi *\Psi }$ in (2.7) can be calculated as follows. For $k=1$, we have that

$$\begin{aligned} \upsilon _{1}(\phi _{1},\psi _{1})=2\phi _{1}*\psi _{1}-\cos \left( \theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}\right) =\cos \left( \theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}\right) . \end{aligned}$$

(3.47)

For $k=2$, we have that

$$\begin{aligned} \upsilon _{2}(\phi _{2},\psi _{2})= & {} \frac{2^{2}\cos (\theta ^{\Phi }_{1})*\phi _{2}*\psi _{2}}{\sin (\theta ^{\Phi }_{1})} -\Big (\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}) \nonumber \\&-\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}) -\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2})\Big ) \nonumber \\= & {} \Big (\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})-\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\Big ) \Big (\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2})+\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2})\Big ) \nonumber \\&-\Big (\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}) -\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}) \nonumber \\&-\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2})\Big ) \nonumber \\= & {} \sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2}). \end{aligned}$$

(3.48)

In general, for an integer $k\in \{2,3,...,2^{n}-1\}$, we have that

$$\begin{aligned} \upsilon _{k}(\phi _{k},\psi _{k})= & {} \frac{2^{k}\left( \prod _{j=1}^{k-1}\cos (\theta ^{\Phi }_{j})\right) *\phi _{k}*\psi _{k}}{\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j})}-\Bigg (\prod _{j=1}^{k-1}\Big (\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Big ) \nonumber \\&\times \Big (\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})+\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big ) -\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big )\Bigg ) \nonumber \\= & {} \prod _{j=1}^{k-1}\Big (\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Big ) \Big (\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})+\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big ) \nonumber \\&-\Bigg (\prod _{j=1}^{k-1}\Big (\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Big ) \nonumber \\&\times \Big (\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})+\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big )-\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big )\Bigg ) \nonumber \\= & {} \prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k}). \end{aligned}$$

(3.49)

Finally, for $k=2^{n}$, we have that

$$\begin{aligned} \upsilon _{2^{n}}(\phi _{2^{n}},\psi _{2^{n}})= & {} \frac{2^{2^{n}-1} \prod _{j=1}^{2^{n}-1}\cos (\theta ^{\Phi }_{j})*\phi _{2^{n}}*\psi _{2^{n}}}{\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j})} -e^{i\left( (\theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi })\right) } \nonumber \\&\times \Bigg (\prod _{j=1}^{2^{n}-1}\Big (\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Big ) -\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})\bigg )\Bigg ) \nonumber \\= & {} e^{i\left( (\theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi })\right) } \Bigg (\prod _{j=1}^{2^{n}-1}\Big (\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Big ) \nonumber \\&-\bigg (\prod _{j=1}^{2^{n}-1}\Big (\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Big ) \nonumber \\&-\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})\bigg )\Bigg ) \nonumber \\= & {} e^{\left( i(\theta _{2^{n}}^{\Phi }+\theta _{2^{n}}^{\Psi })\right) } \prod _{j=1}^{2^{n}-1}\sin \left( \theta ^{\Phi }_{j}+\theta ^{\Psi }_{j}\right) . \end{aligned}$$

(3.50)

Thus, $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi *\Psi }$ in (2.7) with the coefficients in (3.47)–(3.50) satisfies the constraint in (2.2) and has the spherical coordinate $\theta ^{|\Upsilon \rangle ^{\Phi *\Psi }}$ as in (2.8).

Step IV. For the division ($/$) operation, the corresponding coefficients of $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi /\Psi }$ in (2.7) can be calculated as follows. For $k=1$, we have that

$$\begin{aligned} \upsilon _{1}(\phi _{1},\psi _{1})=2(\phi _{1}/\psi _{1})\cos ^{2}(\theta ^{\Psi }_{1})-\cos \left( \theta ^{\Phi }_{1}+\theta ^{\Psi }_{1}\right) =\cos \left( \theta ^{\Phi }_{1}-\theta ^{\Psi }_{1}\right) . \end{aligned}$$

(3.51)

For $k=2$, we have that

$$\begin{aligned} \upsilon _{2}(\phi _{2},\psi _{2})= & {} -\frac{2^{2}\cos (\theta ^{\Phi }_{1})*(\phi _{2}/\psi _{2})*\sin ^{2}(\theta _{1}^{\Psi })\cos ^{2}(\theta ^{\Psi }_{2})}{\sin (\theta ^{\Phi }_{1})}\nonumber \\&+\Big (\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}) +\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2})\nonumber \\&-\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2})\Big )\nonumber \\= & {} -\Big (\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})-\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\Big ) \Big (\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2})+\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2})\Big )\nonumber \\&+\Big (\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}) +\sin (\theta ^{\Phi }_{1}+\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}+\theta ^{\Psi }_{2}) \nonumber \\&-\sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2})\Big ) \nonumber \\= & {} \sin (\theta ^{\Phi }_{1}-\theta ^{\Psi }_{1})\cos (\theta ^{\Phi }_{2}-\theta ^{\Psi }_{2}). \end{aligned}$$

(3.52)

In general, for an integer $k\in \{2,3,...,2^{n}-1\}$, we have that

$$\begin{aligned}&\upsilon _{k}(\phi _{k},\psi _{k})\nonumber \\= & {} -\frac{(-1)^{k-1}2^{k}\left( \prod _{j=1}^{k-1}\cos (\theta ^{\Phi }_{j})\right) *(\phi _{k}/\psi _{k}) *\prod _{j=1}^{k-1}\sin ^{2}(\theta ^{\Psi }_{j})\cos ^{2}(\theta ^{\Psi }_{k})}{\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j})}\nonumber \\&-\Bigg (\prod _{j=1}^{k-1}\Big (-\left( \sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\right) \Big ) \Big (\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})+\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big ) \nonumber \\&-\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})\Big )\Bigg ) \nonumber \\= & {} \prod _{j=1}^{k-1}\Big (-\left( \sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\right) \Big ) \Big (\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})+\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big ) \nonumber \\&-\Bigg (\prod _{j=1}^{k-1}\Big (-\left( \sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\right) \Big ) \Big (\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})+\cos (\theta ^{\Phi }_{k}+\theta ^{\Psi }_{k})\Big ) \nonumber \\&-\prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k})\Big )\Bigg ) \nonumber \\= & {} \prod _{j=1}^{k-1}\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\cos (\theta ^{\Phi }_{k}-\theta ^{\Psi }_{k}). \end{aligned}$$

(3.53)

Finally, for $k=2^{n}$, we have that

$$\begin{aligned} \upsilon _{2^{n}}(\phi _{2^{n}},\psi _{2^{n}})= & {} \frac{(-2)^{2^{n}-1} \prod _{j=1}^{2^{n}-1}\cos (\theta ^{\Phi }_{j})*\phi _{2^{n}}*\psi _{2^{n}}*\prod _{j=1}^{2^{n}-1}\sin ^{2}(\theta ^{\Psi }_{j})\cos ^{2}(\theta ^{\Psi }_{2^{n}})}{\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j})}\nonumber \\&-e^{i\left( (\theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi })\right) } \Bigg (\prod _{j=1}^{2^{n}-1}\Big (-\left( \sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\right) \Big ) \nonumber \\&-\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\bigg )\Bigg ) \nonumber \\= & {} e^{i\left( (\theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi })\right) } \Bigg (\prod _{j=1}^{2^{n}-1}\Big (-\left( \sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\right) \Big ) \nonumber \\&-\prod _{j=1}^{2^{n}-1}\Big (-\left( \sin (\theta ^{\Phi }_{j}+\theta ^{\Psi }_{j})-\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\right) \Big ) \nonumber \\&+\prod _{j=1}^{2^{n}-1}\sin (\theta ^{\Phi }_{j}-\theta ^{\Psi }_{j})\Bigg ) \nonumber \\= & {} e^{\left( i(\theta _{2^{n}}^{\Phi }-\theta _{2^{n}}^{\Psi })\right) } \prod _{j=1}^{2^{n}-1}\sin \left( \theta ^{\Phi }_{j}-\theta ^{\Psi }_{j}\right) . \end{aligned}$$

(3.54)

Thus, $|\Upsilon \rangle (\Phi ,\Psi )=|\Upsilon \rangle ^{\Phi /\Psi }$ in (2.7) with the coefficients in (3.51)–(3.54) satisfies the constraint in (2.2) and has the spherical coordinate $\theta ^{|\Upsilon \rangle ^{\Phi /\Psi }}$ as in (2.8).

Finally, by following from the proofs in Part I and Part II with four proving steps, we can reach a proof for Proposition 2.1. $\square$

Conclusion

In this paper, we derive a general spherical coordinate formula for a quantum state of n-qubit register and study n-qubit operation rules on $(n+1)$-sphere with the target to help developing a (photon or other technique)-based programmable quantum computer. The newly developed angle-based n-qubit operation rules are simple and efficient, which reduce the complicated quantum multiplication and division operations to simple addition and subtraction operations just like those used in a conventional computer. The rules for n-qubit operations are realized through measurement-based feedback controls and quantum entanglements. In the meanwhile, we derive the scaling limits (referred to as RGRFs on the sphere $S^{n+1}$) for n-qubit quantum computer-based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the sphere as arrival inputs. In the first regime, the qubit number n is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit RGRF in terms of reasonably balancing the arrival and service rates under first-in first-out and work-conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number n itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. The second heavy traffic regime is in contrast to the well-known Halfin–Whitt regime where the number of servers is considered as a scaling parameter. Finally, it is worthy to point out that the extension of our current study to a general Riemannian manifold (e.g., a torus corresponding to a scaling quantum computer) is under way.