Mermin polynomials for non-locality and entanglement detection in Grover’s algorithm and Quantum Fourier Transform

Abstract

The non-locality and thus the presence of entanglement of a quantum system can be detected using Mermin polynomials. This gives us a means to study non-locality evolution during the execution of quantum algorithms. We first consider Grover’s quantum search algorithm, noticing that states during the execution of the algorithm reach a maximum for an entanglement measure when close to a predetermined state, which allows us to search for a single optimal Mermin operator and use it to evaluate non-locality through the whole execution of Grover’s algorithm. Then the Quantum Fourier Transform is also studied with Mermin polynomials. A different optimal Mermin operator is searched for at each execution step, since in this case nothing hints us at finding a predetermined state maximally violating the Mermin inequality. The results for the Quantum Fourier Transform are compared to results from a previous study of entanglement with Cayley hyperdeterminant. All our computations can be repeated thanks to a structured and documented open-source code that we provide.

Appendices

Appendix A Explicit states for Grover’s algorithm

Proposition 2

[14, Observation 1] The state $|\varphi _k\rangle$ after k iterations of Grover’s algorithm can be written as follows:

$$\begin{aligned} |\varphi _k\rangle = \tilde{\alpha }_k \sum _{\mathbf{x}\in S} |\mathbf{x}\rangle + \tilde{\beta }_k |+\rangle ^{\otimes n} \end{aligned}$$

(10)

with $\tilde{\alpha }_k=\dfrac{\cos (\frac{2k+1}{2} \theta )}{\sqrt{|S|}} - \dfrac{\sin (\frac{2k + 1}{2}\theta )}{\sqrt{N - |S|}}$ and $\tilde{\beta }_k = 2^{n/2} \dfrac{\sin (\frac{2k + 1}{2}\theta )}{\sqrt{N-|S|}}$.

Proof

With $|\varphi _0\rangle = |+\rangle ^{\otimes n}$, we can write:

$$\begin{aligned} |\varphi _k\rangle = \mathcal {L}^k|\varphi _0\rangle = \dfrac{a_k}{\sqrt{|S|}}\sum _ {\mathbf{x}\in S} |\mathbf{x}\rangle + \dfrac{b_k}{\sqrt{N-|S|}}\sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle \end{aligned}$$

where $\mathcal {L}$ is the loop (oracle and diffusion operator) in Grover’s algorithm.

The oracle is a reflection about $(\sum _{\mathbf{x}\in S}|\mathbf{x}\rangle )^\bot = \sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle$ and the diffusion operator is a reflection about $|+\rangle ^{\otimes n}$. The composition of these two symmetries is a rotation whose angle $\theta$ is the double of the angle between $\sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle$ and $|+\rangle ^{\otimes n}$. So,

$$\begin{aligned} |+\rangle ^{\otimes n}= & {} \frac{1}{\sqrt{|S|}}\sin (\frac{\theta }{2}) \sum _{\mathbf{x}\in S}|\mathbf{x}\rangle +\frac{1}{\sqrt{N-|S|}}\cos (\frac{\theta }{2}) \sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle \\ \frac{1}{\sqrt{N}} \left( \sum _{\mathbf{x}\in S} |\mathbf{x}\rangle +\sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle \right)= & {} \frac{1}{\sqrt{|S|}}\sin (\frac{\theta }{2}) \sum _{\mathbf{x}\in S}|\mathbf{x}\rangle +\frac{1}{\sqrt{N-|S|}}\cos (\frac{\theta }{2}) \sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle \\ \frac{1}{\sqrt{N}}\sum _{\mathbf{x}\in S} |\mathbf{x}\rangle= & {} \frac{1}{\sqrt{|S|}}\sin (\frac{\theta }{2}) \sum _{\mathbf{x}\in S} |\mathbf{x}\rangle \\ \frac{1}{\sqrt{N}}= & {} \frac{1}{\sqrt{|S|}}\sin (\frac{\theta }{2})\\ \sin (\frac{\theta }{2})= & {} \sqrt{\frac{|S|}{N}}. \end{aligned}$$

The fact that $\mathcal {L}$ is a rotation of angle $\theta$ gives $a_k=\sin \left( \theta _k\right)$ and $b_k = \cos \left( \theta _k \right)$ with $\theta _k = k \theta + \theta /2$. Equation (1) then comes from $\alpha _k = \frac{1}{\sqrt{|S|}}\sin (\frac{2k + 1}{2} \theta )$ and $\beta _k = \frac{1}{\sqrt{N-|S|}}\cos (\frac{2k + 1}{2}\theta )$.

With this, we can now take $\tilde{\alpha }_k = \alpha _k - \beta _k$ and $\tilde{\beta }_k = 2^{n/2} \beta _k$ which gives us

$$\begin{aligned} |\varphi _k\rangle= & {} \alpha _k \sum _{\mathbf{x}\in S} |\mathbf{x}\rangle + \beta _k \sum _{\mathbf{x}\notin S} |\mathbf{x}\rangle \\= & {} (\alpha _k - \beta _k) \sum _{\mathbf{x}\in S} |\mathbf{x}\rangle + \beta _k \sum _{\mathbf{x}=0}^{N-1} |\mathbf{x}\rangle \\= & {} \tilde{\alpha }_k\sum _{\mathbf{x}\in S}|\mathbf{x}\rangle + \tilde{\beta }_k |+\rangle ^{\otimes n} \end{aligned}$$

since $|+\rangle ^{\otimes n} = \left( \frac{1}{\sqrt{2}}\right) ^n \sum _{\mathbf{x}=0}^{N-1} |\mathbf{x}\rangle$. $\square$

Proposition 3

In Proposition 2, $\tilde{\alpha }_k$ increases for k between 0 and $\frac{\pi }{4}\sqrt{\frac{N}{|S|}}-\frac{1}{2}$ and $\tilde{\beta }_k$ decreases on the same interval.

Proof

The optimal number of iterations of the loop $\mathcal {L}$ in Grover’s algorithm is the smallest value $k_{opt}$ of k such that $a_k = 1$, i.e., $\theta _{k_{opt}} = \pi /2$. With $|S|\ll N$, $\sin \left( \theta /2\right) =\sqrt{|S|/N}$ gives $\theta \approx 2\sqrt{|S|/N}$ and $\theta _k \approx (2k+1)\sqrt{|S|/N}$. Finally $(2k_{opt}+1)\sqrt{|S|/N}$ optimally approximates $\pi /2$ if $k_{opt} = \left\lfloor \frac{\pi }{4}\sqrt{\frac{N}{|S|}}-\frac{1}{2} \right\rceil = \left\lfloor \frac{\pi }{4}\sqrt{\frac{N}{|S|}} \right\rfloor$.

Moreover, $a_k=\sin \left( \theta _k\right)$ and $\alpha _k = \frac{1}{\sqrt{|S|}} a_k$ are increasing and $b_k = \cos \left( \theta _k \right)$ and $\beta _k = \frac{1}{\sqrt{N-|S|}} b_k$ are decreasing for k from 0 to $\left( \frac{\pi }{4}\sqrt{\frac{N}{|S|}}-\frac{1}{2}\right)$. From the expressions $\tilde{\alpha }_k = \alpha _k - \beta _k$ and $\tilde{\beta }_k = 2^{n/2} \beta _k$, we get the result of the proposition.$\square$

Appendix B Cayley hyperdeterminant $\varDelta _{2222}$

Let $|\varphi \rangle =\sum _{i,j,k,l\in \{0,1\}} a_{i,j,k,l}|ijkl\rangle$ be a four-qubit state. The algebra of polynomial invariants for the four-qubit Hilbert space can be generated by the four polynomials H, L, M and D defined as follows [21]:

$$\begin{aligned} H= & {} a_{0000}a_{1111} - a_{1000}a_{0111} - a_{0100}a_{1011} + a_{1100}a_{0011}\\&-a_{0010}a_{1101} + a_{1010}a_{0101} + a_{0110}a_{1001} - a_{1110}a_{0001} \end{aligned}$$

is an invariant of degree 2.

$$\begin{aligned} L=\left| \begin{array}{cccc} a_{0000}&{}a_{0010}&{}a_{0001}&{}a_{0011}\\ a_{1000}&{}a_{1010}&{}a_{1001}&{}a_{1011}\\ a_{0100}&{}a_{0110}&{}a_{0101}&{}a_{0111}\\ a_{1100}&{}a_{1110}&{}a_{1101}&{}a_{1111} \end{array}\right| \quad \text{ and } M=\left| \begin{array}{cccc} a_{0000}&{}a_{0001}&{}a_{0100}&{}a_{0101}\\ a_{1000}&{}a_{1001}&{}a_{1100}&{}a_{1101}\\ a_{0010}&{}a_{0011}&{}a_{0110}&{}a_{0111}\\ a_{1010}&{}a_{1011}&{}a_{1110}&{}a_{1111} \end{array}\right| \end{aligned}$$

are two invariants of degree 4.

Consider the partial derivative

$$\begin{aligned} b_{xt}:=\det \left( \dfrac{\partial ^2 A}{\partial y_i\partial z_j}\right) \end{aligned}$$

of the quadrilinear form $A =\sum _{i,j,k,l\in \{0,1\}} a_{i,j,k,l} x_i y_j z_k t_l$ with respect to the variables y and z. This quadratic form with variables x and t can be interpreted as a bilinear form on the three-dimensional space $\text {Sym}^2(\mathbb {C}^2)$, i.e., there is a $3\times 3$ matrix $B_{xt}$ satisfying

$$\begin{aligned} b_{xt}=[x_0^2,x_0x_1,x_1^2]~B_{xt}~\left[ \begin{array}{c} t_0^2\\ t_0t_1\\ t_1^2 \end{array}\right] . \end{aligned}$$

Then $D =\det (B_{xt})$ is an invariant of degree 6.

Let’s introduce the invariant polynomials

$$\begin{aligned} U= & {} H^2-4(L-M), \quad V=12(HD-2LM),\\ S= & {} \dfrac{1}{12}(U^2-2V) \quad \text{ and } \quad T=\dfrac{1}{216}(U^3-3UV+216D^2). \end{aligned}$$

Then the Cayley hyperdeterminant is [21]:

$$\begin{aligned} \varDelta _{2222}=S^3-27T^2. \end{aligned}$$