# On Unitary t-Designs from Relaxed Seeds

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## Abstract

**:**

## 1. Introduction and Summary of the Results

#### 1.1. Unitary t-Designs

**Definition**

**1.**

#### 1.2. Comparison with Previous Work

- Requirement (i): every $U\in {\mathcal{U}}_{\mathcal{B}}$ has an inverse ${U}^{\u2020}\in {\mathcal{U}}_{\mathcal{B}}$.
- Requirement (ii): the unitaries $U\in {\mathcal{U}}_{\mathcal{B}}$ are composed entirely of algebraic entries.

#### 1.3. Main Results

**Theorem**

**1.**

**Theorem**

**2.**

**Definition**

**2.**

**Proposition**

**1.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

#### 1.4. Example: Implementation of Our Construction as a Random Quantum Circuit

## 2. Proofs

#### 2.1. Proof of Theorem (2)

- Without loss of generality, let ${\mathcal{U}}_{\mathcal{M}}=\{{W}_{1},\dots ,{W}_{n}\}$ and ${\mathcal{U}}_{\mathcal{B}/\mathcal{M}}=\{{V}_{1},\dots ,{V}_{m}\}$, with $m,n\in \mathbb{N}$; and let ${V}_{{j}_{1}}={V}_{m}$. Fix $\{{W}_{1},\dots ,{W}_{n},{V}_{1},\dots ,{V}_{m-1}\}$, and list all the possible relations of the form of the right-hand side of Equation (23), where ${W}_{j}\in \{{W}_{1},\dots ,{W}_{n}\}$, $\forall j\in \{k+1,\dots ,2k\}$, and ${V}_{i},{V}_{j}\in \{{V}_{1},\dots ,{V}_{m-1}\}$, $\forall i\in \{1,\dots ,k\}$, $\forall j\in \{k+1,\dots ,{j}_{1}-1,{j}_{1}+1,\dots 2k\}$. Since there are $countably$ many relations of the form of the right-hand side of Equation (23) (and $uncountably$ many choices of ${V}_{m}$.), choose ${V}_{{j}_{1}}={V}_{m}$ such that it is not equal to any of the listed relations of the right-hand side of Equation (23). Therefore, Equation (23) does not hold in general in
**Case 1**. - Here, it will be convenient to rewrite Equation (23) as$${V}_{{j}_{1}}=\prod _{i=1,\dots ,2k-1}{C}_{i}^{\pi \left(i\right)}{\left({V}_{{j}_{1}}^{\u2020}\right)}^{1-\pi \left(i\right)},$$
**Case 1**). $\pi (.)$ is a map$$i=\{1,\dots ,2k-1\}\to \pi \left(i\right)\in \{0,1\}.$$We consider the two following subcases **Case 2a:**$\pi \left(i\right)=0$, $\forall i\in \{1,\dots ,2k-1\}.$Equation (24) becomes, in this case,$${V}_{{j}_{1}}={\left({V}_{{j}_{1}}^{\u2020}\right)}^{2k-1}.$$**Case 2b:**∃${i}_{1}$ such that $\pi \left({i}_{1}\right)=1$.Equation (24) can be rewritten in this case as$${C}_{{i}_{1}}=\prod _{i={i}_{1}-1,\dots ,1}{V}_{{j}_{1}}^{1-\pi \left(i\right)}{C}_{i}^{\u2020\pi \left(i\right)}{V}_{{j}_{1}}\prod _{i=2k,\dots ,{i}_{1}+1}{V}_{{j}_{1}}^{1-\pi \left(i\right)}{C}_{i}^{\u2020\pi \left(i\right)}.$$Since ${C}_{{i}_{1}}\in \{{V}_{1}^{\u2020},\dots ,{V}_{m-1}^{\u2020},{W}_{1}^{\u2020},\dots ,{W}_{n}^{\u2020}\}$, and these unitaries are fixed, Equation (26) therefore cannot hold for a general choice of ${V}_{{j}_{1}}={V}_{m}$.In order to complete the proof of Theorem (2), we should show that a ${V}_{m}$ exists which simultaneously violates the relations imposed in**Case 1**and**Case 2**. For a given fixed integer k and fixed $\{{W}_{1},\dots ,{W}_{n},{V}_{1},\dots ,{V}_{m-1}\}$, there is only a finite number of unitaries ${V}_{m}$ satisfying Equation (23) in**Case 1**. Unitaries ${V}_{m}$ satisfying Equations (25) and (26) (**Case 2a**and**2b**) also satisfy the relation$$det({C}_{{i}_{1}}-\prod _{i={i}_{1}-1,\dots ,1}{V}_{{j}_{1}}^{1-\pi \left(i\right)}{C}_{i}^{\u2020\pi \left(i\right)}{V}_{{j}_{1}}\prod _{i=2k,\dots ,{i}_{1}+1}{V}_{{j}_{1}}^{1-\pi \left(i\right)}{C}_{i}^{\u2020\pi \left(i\right)})=0.$$Using the analysis of [27], the set of unitaries ${V}_{m}$ satisfying relations of the form Equation (27) has zero Haar measure on U(4). This follows from the fact that one can show that there is a one-to-one mapping between these (nonidentically zero) polynomial equations in the matrix elements of ${V}_{m}$, and the intersection (Corresponding to partitioning the determinant into real and imaginary parts, each of which can be expressed as a trigonometric function of 16 real valued angles in $[0,2\pi ]$ parametrizing ${V}_{m}$[27].) of the zero sets of two real analytic functions on ${\mathbb{R}}^{16}$. Each such zero set has a Lebesgue measure zero, therefore, their intersection (which is a subset of the two) also has Lebesgue measure zero (see [27] for more details). Therefore, the set of unitaries generated by relations of the form of Equation (27) has Haar measure zero [27]. The number of possible relations of the form of Equation (27) is countable (for fixed k and fixed $\{{W}_{1},\dots ,{W}_{n},{V}_{1},\dots ,{V}_{m-1}\}$), thus the Haar measure of the set of unitaries ${V}_{m}$ satisfying Equations (25) or (26) is also zero, as the countable union of measure zero sets is also measure zero. This means that we can choose ${V}_{m}$ to be outside a measure zero set (which is the set of unitaries satisfying Equations (23) in**Case 1**, (25), and (26)), and we would therefore have that ${V}_{m}$ simultaneously violates the relations imposed by**Case 1**and**Case 2**. This completes the proof of Theorem (2).

#### 2.2. Proof of Theorem (3)

#### 2.3. Proof of Theorem (4)

#### 2.4. Proof of Theorem (5)

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Part of the random quantum circuit sampling from the random unitary ensemble $bloc{k}^{L}\left({B}_{1}\right)$. The horizontal black lines numbered from 1 to n represent the n input qubits of the random quantum circuit. The colored boxes touching two horizontal lines each represent a two-qubit unitary which is chosen with uniform probability from ${\mathcal{U}}^{k}$ (Equation (11)). These two-qubit unitaries act nontrivially only on the horizontal lines (qubits) they touch. The order in which these unitaries are applied is from left to right. Unitaries (boxes) aligned on the same vertical level are applied simultaneously (depth-one). The depth-two unitary shown in this figure is sampled from $block\left({B}_{1}\right)$. In order to sample from $bloc{k}^{L}\left({B}_{1}\right)$, the $\epsilon $-approximate t-design, the random circuit shown in this figure is repeated L times, with L given by Equation (17) (see also Theorem (5)). This figure is for n-even, the odd n case follows straightforwardly.

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Mezher, R.; Ghalbouni, J.; Dgheim, J.; Markham, D.
On Unitary *t*-Designs from Relaxed Seeds. *Entropy* **2020**, *22*, 92.
https://doi.org/10.3390/e22010092

**AMA Style**

Mezher R, Ghalbouni J, Dgheim J, Markham D.
On Unitary *t*-Designs from Relaxed Seeds. *Entropy*. 2020; 22(1):92.
https://doi.org/10.3390/e22010092

**Chicago/Turabian Style**

Mezher, Rawad, Joe Ghalbouni, Joseph Dgheim, and Damian Markham.
2020. "On Unitary *t*-Designs from Relaxed Seeds" *Entropy* 22, no. 1: 92.
https://doi.org/10.3390/e22010092