# Visualizing Quantum Circuit Probability: Estimating Quantum State Complexity for Quantum Program Synthesis

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## Abstract

**:**

## 1. Introduction

## 2. States and Complexities

#### 2.1. The Statistical Emergence of Entropy

#### 2.2. The Algorithmic Emergence of Universality

#### 2.3. Relations to Circuit Complexity

- 1.
- Statistical complexity: Shannon entropy on an ensemble of states (given its probability distribution)
- 2.
- Computational complexity: Space-time scaling behavior of a program to generate the state (given a language)
- 3.
- Algorithmic complexity: Length of the program to generate the state (given a language)

## 3. Landscape of Circuits

#### 3.1. Circuit Probability of States

#### 3.2. Boolean Gate Sets

- For 1-input Boolean algebra, i.e., when $v=1$, $s=2$, $d=2$, the total number of functions are $f={2}^{{2}^{1}}=4$. These functions are the $\{0,1,A,\overline{A}\}$.
- For 2-input Boolean algebra, i.e., when $v=2$, $s=2$, $d=2$, the total number of functions are $f={2}^{{2}^{2}}=16$. These are denoted by $\{0,1,A,B,\overline{A},\overline{B},A\u2022B,\overline{A\u2022B},A+B,\overline{A+B},A+\overline{B},\overline{A}+B,A\u2022\overline{B},\overline{A}\u2022B,A\oplus B,\overline{A\oplus B}\}$.

`NAND`}, {

`NOR`}, {

`NOT`,

`AND`}, {

`NOT`,

`OR`}. These gate sets are related to each other, using the following equivalences:

`NOT(A) = NAND(A,A) = NOR(A,A)`,`OR(A,B) = NAND(NAND(A,A),NAND(B,B)) = NOR(NOR(A,B),NOR(A,B))``= NOT(AND(NOT(A),NOT(B)))`,`AND(A,B) = NAND(NAND(A,B),NAND(A,B)) = NOR(NOR(A,A),NOR(B,B))``= NOT(OR(NOT(A),NOT(B)))`.

#### 3.3. Quantum Gate Sets

`CCX`gate (also called the Toffoli gate). It can simulate the

`NAND`gate via

`CCX(A,B,1) = (A,B,NAND(A,B))`. Classical computations are in general irreversible processes, thus, the inputs cannot be recovered from the outputs. Quantum logic is based on unitary evolution and thus, is reversible. Additionally, quantum computations allow quantum superposition and entanglement, which are not implied in reversible computation. The

`CCX`gate can simulate the entire set of reversible computations by additionally simulating a

`Fanout`gate (or Copy gate) as

`CCX(A,1,0) = (A,1,A)`. Thus, both {

`NAND`,

`Fanout`} and {

`CCX`} form universal gate sets for reversible computation. The

`CSWAP`gate (also called the Fredkin gate) is another universal gate for reversible logic.

`H`gate (Hadamard). In principle, the real gate set composed of {

`CCX`,

`H`} is computationally universal [42]. However, it needs ancilla qubits to encode the real normalization factors and complex algebra to decompose [43] to arbitrary quantum unitary gates for a strong sense of universality. Moreover, it is important that the effect of the

`NOT`gate (or, the

`X`gate in quantum) cannot be simulated without assuming the availability of both $|0\rangle $ and $|1\rangle $ states. Since our enumeration of quantum programs will start will the qubits initialized to the all-zero state, we need to augment the gate set to {

`X`,

`H`,

`CCX`} to reach all binary strings as output.

`RY`$\left(\theta \right)$,

`RZ`$\left(\theta \right)$,

`CX`}. It can be recursively applied to larger quantum circuits with the CX count scaling of $O\left({4}^{n}\right)$.

`H`,

`T`,

`CX`}), runs in $O\left(log\right(1/\u03f5\left)\right)$ time, and produces as output a sequence of $O\left(log\right(1/\u03f5\left)\right)$ quantum gates which approximate the desired quantum gate to an accuracy within $\u03f5>0$. It can be generalized to apply to multi-qubit gates and to gates from $SU\left(d\right)$.

## 4. Implementation

#### 4.1. Gate Sets

- 1.
- {
`CCX`}—This set is universal for classical and reversible logic, provided both the initial states of $|0\rangle $ and $|1\rangle $ are provided. It is not practical to provide all initial states without knowing how to create one from the other. Since all gate-based quantum algorithms start from the all-$|0\rangle $ state and prepare the required initial state via gates, we will not consider this set for our enumeration. - 2.
- {
`X`,`CCX`}—This set is universal for classical and reversible logic by starting from the all-$|0\rangle $ state. - 3.
- {
`X`,`H`,`CCX`}—This set is weakly universal under encoding and ancilla assumptions for quantum logic. The encoding, while universal, might not preserve the computation resource complexity benefits of quantum (i.e., in the same way, classical computation can also encode all quantum computation using {`NAND`,`Fanout`}). Thus, we do not consider this set for our enumeration of the quantum case. - 4.
- {
`H`,`S`,`CX`}—The Clifford group is useful for quantum error correction. However, it is non-universal and can be efficiently simulated on classical logic [52]. The space of transforms on this set encoded error-correction codes and is, thus, useful to map. - 5.
- {
`H`,`T`}—This set is universal for single qubit quantum logic. However, we will consider the generalization to multi-qubit using an additional two-qubit gate in the set in the following case. - 6.
- {
`H`,`T`,`CX`}—This is universal for quantum logic. - 7.
- {
`P(pi/4)`,`RX(pi/2)`,`CX`}—The IBM native gate set is used to construct this gate set. The following relations establish the relation with the previous universal gate set: $\mathtt{T}=\mathtt{P}(\mathtt{p}\mathtt{i}/\mathtt{4})$, $\mathtt{X}=\mathtt{R}\mathtt{X}(\mathtt{p}\mathtt{i}/\mathtt{2})$, and, $\mathtt{H}={e}^{i\xdf/2}\mathtt{X}\mathtt{R}\mathtt{z}(\mathtt{p}\mathtt{i}/\mathtt{2})\mathtt{X}={e}^{i\xdf/2}\mathtt{X}\mathtt{T}\mathtt{T}\mathtt{T}\mathtt{T}\mathtt{X}$. We will consider additional constraints like device connectivity to apply this technique to real quantum processors.

`X`,

`CCX`}, (ii) {

`H`,

`S`,

`CX`}, (iii) {

`H`,

`T`,

`CX`}, and (iv) {

`P(pi/4)`,

`RX(pi/2)`,

`CX`}.

#### 4.2. Metrics for Evaluation

- Expressivity: refers to the extent to which the Hilbert space can be encoded by using an unbounded number of gates. It is not weighted by the probability as it is a characteristic of the encoding power of the gate set. We assign a 1 to a final state if it can be expressed as starting from the initial state and applying a sequence of gates from the gate set.
- Reachability: refers to a bounded form of expressibility. The length of the sequence of gates must be equal to or shorter than the specified bound. This corresponds to a physical implementation rather than the power of the gate set and characterizes the computational complexity and thereby the decoherence time of the processor.

#### 4.3. Enumeration Procedure

`X`,

`CCX`}), this corresponds to the statistics of the number of computational paths between these two states using an arrangement of gates from the set, conforming to a specified length. For the quantum case, the statistics correspond to the sum of probabilities of the computational paths collapsing on measurement to the target state. Dividing the matrix by the total number of programs gives us the fixed-length algorithmic probability of the state on each row, conditioned on the initial state. This normalized $n\times n$ matrix is the reachability landscape. All non-zero values correspond to the states that are reachable by at least one route (i.e., at least one program exists to transform to that state). This gives us the Boolean $n\times n$ expressibility matrix.

#### 4.4. Results

#### 4.5. Analysis and Discussion

## 5. Applications

#### 5.1. Geometric Quantum Machine Learning

#### 5.2. Novel Quantum Algorithm Synthesis

#### 5.3. Quantum Artificial General Intelligence

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Growth of the number of programs with qubit count and circuit depth for two types of gate sets: (i) $[1,3]$ qubits: $\{\mathtt{X},\mathtt{CCX}\}$, (ii) $[1,1,2]$ qubits: $\{\mathtt{H},\mathtt{S},\mathtt{CX}\}$, $\{\mathtt{H},\mathtt{T},\mathtt{CX}\}$, $\left\{\mathtt{P}\right(\xdf/\mathtt{4}),\mathtt{RX}(\xdf/\mathtt{2}),\mathtt{CX}\}$.

**Figure 2.**Expressibility and Reachability for gate set $\{\mathtt{X},\mathtt{CCX}\}$ on 4 qubits and of circuit depth from 0 to 3.

**Figure 3.**Expressibility and Reachability for gate set $\{\mathtt{H},\mathtt{S},\mathtt{CX}\}$ on 4 qubits and of circuit depth from 0 to 3.

**Figure 4.**Expressibility and Reachability for gate set $\{\mathtt{H},\mathtt{T},\mathtt{CX}\}$ on 4 qubits and of circuit depth from 0 to 3.

**Figure 5.**Expressibility and Reachability for gate set $\left\{\mathtt{P}\right(\xdf/\mathtt{4}),\mathtt{RX}(\xdf/\mathtt{2}),\mathtt{CX}\}$ on 4 qubits and of circuit depth from 0 to 3.

**Figure 6.**Approximation of circuit probability of states on 4 qubits for $L=3$ using two gate sets (

**a**) $\{\mathtt{X},\phantom{\rule{3.33333pt}{0ex}}\mathtt{CCX}\}$ and (

**b**) $\{\mathtt{H},\phantom{\rule{3.33333pt}{0ex}}\mathtt{T},\phantom{\rule{3.33333pt}{0ex}}\mathtt{CX}\}$.

**Figure 7.**Expressibility and Reachability for gate set $\left\{\mathtt{P}\right(\xdf/\mathtt{4}),\mathtt{RX}(\xdf/\mathtt{2}),\mathtt{CX}\}$ on 5 qubits and of circuit depth from 0 to 3 on the IBM T-topology.

**Figure 8.**Expressibility and Reachability for gate set $\left\{\mathtt{P}\right(\xdf/\mathtt{4}),\mathtt{RX}(\xdf/\mathtt{2}),\mathtt{CX}\}$ on 5 qubits and of circuit depth from 0 to 3 on the IBM L-topology.

**Figure 9.**${M}_{circ}$ and their comparison for gate set $\left\{\mathtt{P}\right(\xdf/\mathtt{4}),\mathtt{RX}(\xdf/\mathtt{2}),\mathtt{CX}\}$ on 5 qubits and of circuit depth from 0 to 3 on the IBM L-topology and T-topology.

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## Share and Cite

**MDPI and ACS Style**

Bach, B.G.; Kundu, A.; Acharya, T.; Sarkar, A.
Visualizing Quantum Circuit Probability: Estimating Quantum State Complexity for Quantum Program Synthesis. *Entropy* **2023**, *25*, 763.
https://doi.org/10.3390/e25050763

**AMA Style**

Bach BG, Kundu A, Acharya T, Sarkar A.
Visualizing Quantum Circuit Probability: Estimating Quantum State Complexity for Quantum Program Synthesis. *Entropy*. 2023; 25(5):763.
https://doi.org/10.3390/e25050763

**Chicago/Turabian Style**

Bach, Bao Gia, Akash Kundu, Tamal Acharya, and Aritra Sarkar.
2023. "Visualizing Quantum Circuit Probability: Estimating Quantum State Complexity for Quantum Program Synthesis" *Entropy* 25, no. 5: 763.
https://doi.org/10.3390/e25050763