Questions tagged [gate-synthesis]

For questions about finding (short) gate sequences to implement a specific unitary operation, for example decomposing a complicated multi-qubit gate into a sequence of basic gates. It might apply to optimizing circuits with respect to length or depth or finding gate sequences to implement an algorithm.

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18 views

Do we need to use an ancillary qubit when decomposing arbitrary $U(2^n)$ gates using Clifford+T universal gate sets?

As I know, we can decompose $U$ without ancilla if it's from special unitary group $SU(2^n)$. Do we need to use ancilla qubit on decomposing arbitrary $n$-qubit $U$ using Clifford+T universal gates ...
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2answers
388 views

Why is the decomposition of a qubit-qutrit Hamiltonian in terms of Pauli and Gell-Mann matrices not unique?

If I have the $X$ gate acting on a qubit and the $\lambda_6$ gate acting on a qutrit, where $\lambda_6$ is a Gell-Mann matrix, the system is subjected to the Hamiltonian: $\lambda_6X= \begin{pmatrix}0 ...
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1answer
64 views

What is Qiskit's Transpiler method for unitary synthesis?

As I could found in here how the transpile works in qiskit, I understood that transpile gets arbitrary Unitary gate $U$ and some set of basis gates as input, and produce some quantum circuit of $U$ ...
3
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1answer
75 views

How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?

From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by $$ R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
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1answer
69 views

IBM Qiskit QAOA gate implementation question

In section $5.2$ of the QAOA chapter in Qiskit textbook, section $5.2$, state preparation uses the gate $U_{k,l}(\gamma) = e^{\frac{i \gamma}{2} (1-Z_k Z_l)}$. Later, in section $5.3$, this gate is ...
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1answer
213 views

Is it possible to express $U_1(\lambda)$ through the gates $R_x, R_y, R_z$ while maintaining the phase? In Qiskit for example

Is it possible to express gate $U_1(\lambda)$ through the gates $R_x, R_y, R_z$ while maintaining the phase? Both in principle and in practice (in Qiskit for example)? The single gate $R_z(\lambda)$ ...
4
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1answer
97 views

Single-qubit rotations on a subspace within two-qubit unitary

I would like to implement the operation $$ U(a,b) = \exp\left(i \frac{a}{2} (XX + YY) + i \frac{b}{2} (XY - YX) \right) $$ ($a,b \in \mathbb{R}$) without using Baker-Campbell-Hausdorf expansion, ...
3
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1answer
51 views

From mathematical notation to quantum circuit, in general

I am learning the basics of quantum computing using Qiskit and I encountered a problem when I tried to solve some of our course exercises. I feel like I am missing an invisible step, the step from ...
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3answers
100 views

Decomposing gates resembling exponentiated members of desired gateset

Suppose I have access to a pretty typical gate set, for example $\{\text{CNOT}, \text{SWAP}, \text{R}_{x}, \text{R}_{y}, \text{R}_{z}, \text{CR}_x, \text{CR}_y, \text{CR}_z\}$ where $\text{CR}$ is a ...
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3answers
111 views

Find unitary such that $U:|i\rangle|0\rangle\rightarrow|i\rangle|A_i\rangle$

Let's assume I have two qubits of state $|A_0\rangle$ and $|A_1\rangle$ correspondingly stored in a quantum memory. How do I find a Unitary $U$ that acts on another register of 2-qubits such that $$U:|...
3
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1answer
127 views

How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?

How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift? I am stuck in this and any help would be appreciated.
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6answers
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How do I build a gate from a matrix on Qiskit?

I'm creating a gate for a project and need to test if it has the same results as the original circuit in a simulator, how do I build this gate on Qiskit? It's a 3 qubit gate, 8x8 matrix: $$ \frac{1}{...
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1answer
41 views

How does MCPhaseGate/MCU1Gate works internally in qiskit?

I was curious about the implementation of MCPhase/MCU1Gate and how it works without ancilla qubits. I ended up checking the code of the some auxiliary (?) function ...
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2answers
60 views

How to implement $\sqrt{iSWAP}$ in Qiskit

I want to implement the $\sqrt{iSWAP}$ operator using simple operations in Qiskit such as it is done for the $iSWAP$ here or $\sqrt{SWAP}$ gate here. How can I do this? If possible I would like to ...
6
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1answer
174 views

More efficient implementation of $4$-qubit gate

While working on an error detection algorithm, I stumbled upon the problem of simplifying the following implementation Here, the $S$ gate is defined by $$S=\left( \begin{array}{cc} \frac{\sqrt{3}}{2}...
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1answer
179 views

Sampling random circuits vs Solovay-Kitaev compiler

Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix. The gate set is represented by a discrete universal set, e.g. Clifford+T gates or $\{T,H\}$ gates. A well ...
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16 views

Compiling Quantum Circuits using the Palindrome Transform

This paper shows a way to produce optimal circuits. I haver verified most of them and they are correct except this procedure: procedure ProduceArray(n) I cannot ...
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0answers
41 views

Representation of multiple qubit matrices in Dirac notation

Imagine one wants to represent the and function for any number of qubits in Dirac notation. The and gate flips the target qubit if all the control qubits are in state 1. This is its matrix ...
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3answers
99 views

How to create the state $\vert 0 \rangle+i \vert 1 \rangle$ using elementary gates?

I am trying to write $|0\rangle+i|1\rangle$ in terms of elementary gates like H, CNOT, Pauli Y, using the IBM QE circuit composer. I was thinking some kind of combination of H and Y since $Y|0\rangle=...
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1answer
96 views

Gate-level implementation of Eigenvalue-Inversion in HHL

I am trying to understand how does the gate-level implementation of eigenvalue-inversion step in the HHL algorithm works. I am following this reference, where it is stated (Lemma 4) that this can be ...
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1answer
34 views

How to create a gate with functionality CCX(a,b,b)?

Can we create a Controlled gate with below functionality? if {a==|1> && b==|1>} then {qc.x(b)} Basically, a CCX gate but the output Qubit is actually one of the input Qubits. Apparently, ...
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2answers
824 views

What components are needed to realize a photonic CNOT gate?

In Realization of a photonic CNOT gate sufficient for quantum computation FIG. 1 there is a "scheme to obtain a photonic realization of a CNOT gate with two independent qubits." What ...
5
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1answer
191 views

Decomposition of any 2-level matrix into single qubit and CNOT gates

I saw an example which takes a 2 level matrix. Which is a $8\times8$ matrix that acts non trivially only on 2 levels of only states $|000\rangle$ and $|111\rangle$. The way they do it is by using a ...
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1answer
98 views

How to define Q-operator in Quantum Amplitude Estimation

I'm trying to implement a circuit for Quantum Amplitude Estimation in Qiskit using elementary gates. I have created the circuit that represent my algorithm $A$ but now from the theory I know that I ...
3
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1answer
105 views

What are the differences between the Toffoli and Fredkin gates (historical, practical, etc.)

I'm trying to understand the historical ordering and the practical differences between the Toffoli Gate and the Fredkin Gate. Toffoli's February 1980 tech report MIT/LCS/TM-151 states: Where ...
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0answers
50 views

Reducing the depth of quantum circuits with ancilla qubits

This question is two-fold and considers general $n$-qubit operations on a quantum computer. First, can a general $n$-qubit operation be implemented on a quantum computer without the use of ancilla ...
4
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1answer
161 views

CNOT expressed with CZ and H gates by taking into account HZH =X

From this link: Where equation 1 is: I can probably brute-force this by explicitly calculating this quantum circuit's effective 4x4 matrix and seeing that its equivalent to this teleportation ...
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1answer
194 views

How to code a projector operator in qiskit?

I'm new to qiskit and I want to know how do I define a projector operator in qiskit? Specifically, I have prepared a 3 qubit system, and after applying a whole lot of gates and measuring it in a state ...
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2answers
101 views

Decomposition of $|110\rangle \leftrightarrow |000\rangle$ Exchange Gate

How to implement a 3 qubit gate, that exchanges the level $|110\rangle$ and $|000\rangle$, with elementary gates (CNOT, SWAP, Toffoli, local gates, etc.(everything Qiskit allows)): $$ U=\pmatrix{ 0 &...
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1answer
65 views

Principal square root of Pauli Y gate in Qiskit?

I've seen a similar question asked (How do I compute the square root of the $Y$ gate?) but I'm trying to understand how I can use the gates $Y^{\frac{1}{2}}$ or $Y^{\frac{1}{4}}$ in Qiskit in terms of ...
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4answers
3k views

How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?

In a three-qubit system, it's easy to derive the CNOT operator when the control & target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in ...
4
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1answer
68 views

Adding a phase to qubit: why is it necessary for arbitrary single qubit gate

An arbitrary single qubit gate can be decomposed as: $$U=e^{i \alpha} R_z(\beta) R_y(\gamma) R_z(\delta)$$ We notice that in addition to the three rotations, there is a coefficient $e^{i \alpha}$. ...
4
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1answer
153 views

How can you decompose Grover's diffusion operator into gates?

I know how Grover's diffusion operator works ($U_s = 2|s\rangle\langle s|-I$) with the inversion around the mean. However, I want to implement it in simpler gates, to use the algorithm. How can I do ...
7
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1answer
1k views

Would IBM's “compiler” turn my identity circuit into nothing?

If I were to create a circuit with the following gate: $$\tag{1}R_\phi = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \phi} \end{bmatrix},$$ with $\phi$ specified to be equal to 0, then the gate that I ...
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2answers
236 views

How does the stated Pauli decomposition for $\operatorname{CP\cdot A\cdot CP}$ arise?

I'm having a bit of trouble understand @DaftWullie's answer here. I understood that the $4\times 4$ matrix $A$ $$ \frac{1}{4} \left[\begin{matrix} 15 & 9 & 5 & -3 \\ 9 & 15 & 3 &...
5
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0answers
38 views

Equivalence checking of quantum circuits up to error

Suppose you are given two circuit descriptions $A$ and $B$ where by a circuit description I mean a sequence of gates (in the order they are applied) and the qubits they are applied on. (For the sake ...
4
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2answers
159 views

How to find a circuit for the roots of QFT?

After reading about using quantum gates instead of ancillas, it asserts that every quantum circuit has a square root. Theoretically, they do, but is there a practical method to generate the quantum ...
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1answer
92 views

Efficient implementation of exponential of projector

If I have an $n$ qubit system and a projector $P$ such as $P_0 = \left|0\right>^{\otimes n}\left<0\right|^{\otimes n}$ (as an example) on those qubits, is there an efficient way to implement the ...
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0answers
35 views

[CNOT GATE]how to go from a passage matrix acting on |C0>,C1>|T0>,T1> to the Cnot matrix acting on |C0T0>,C0T1>

In article A controlled-NOT gate for frequency-bin qubits, the authors built a passage matrix acting on the states $T_0$, $T_1$, $C_0$, $C_1$ and then they infered the famous two qubit matrix notation....
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2answers
122 views

Are Toffoli gates actually used in designing quantum circuits?

In an actual quantum computer, are we designing circuits with Toffoli Gates and then using compilers or optimizers to remove redundancies so that we can use fewer qubits than a full Toffoli gates ...
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2answers
123 views

How to construct a CU3 gate using only CX and U3 gates?

Knowing that CX and U3 (taking 3 parameters $\theta, \phi$ and $\lambda$) form a set of universal gates how can I construct an arbitrary CU3 gate using a decomposition of only CX and arbitrary U3 ...
3
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3answers
143 views

What is the complexity of splitting a state into a superposition of $n$ computational basis states?

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ I'...
3
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1answer
70 views

Implementing a controlled-controlled-U using controlled-U

Suppose I know how to implement a 2 qubit gate $C-U$ (i.e controlled U), and I want to implement $CC-U$ using $C-U$ and other 1 or 2 qubit gates, is that possible?
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1answer
70 views

Cheap Toffoli gates with phase errors

Here, a cheap verion of a Toffoli, up to a phase flip for $|101\rangle$, is given by with $A=R_y(\pi/4)$. Are there similar versions of cheap implementation of general $C^nNOT$ gates? I tried to ...
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0answers
44 views

How can I prove the universality of this set of gates?

I was reading this article. A brief explanation: Here we have a circuit, the registers are a stepfunction state, an single photon state and a function state. The first two have position operators $X$ ...
2
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0answers
69 views

Decompose Toffoli gate with minimum cost for IBM quantum computer

The known decomposition of toffoli gate that can be used on IBM quantum computer is : I want to know any other Toffoli gate decompositions that can be used on IBM quantum computer and have a cost ...
5
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2answers
328 views

How can I fill a unitary knowing only its first column?

I have a unitary matrix that I want to construct. I only care what happens to the first computational state, so the first column is specified. So far, I've been assigning each question mark to a ...
2
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2answers
105 views

How to prove that a matrix is an arbitrary unitary?

My goal is to prove that I can synthesise arbitrary unitary from two components. In the end, I find a matrix with the form \begin{equation} \mathbf{W}_j=\begin{pmatrix} |\alpha|2\cos{(\phi_{...
3
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0answers
47 views

Cost of controlled-$U_i$

What is the cost (number of gates) of $\sum_{i=0}^{N-1}| i \rangle \langle i|\otimes U_i$ in terms of $N$ and the costs of the unitaries $U_i$? Say the gate set consists of arbitrary one-qubit gates ...
9
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3answers
1k views

How can I see, without math, the action of a gate in matrix form?

Suppose we have the Fredkin gate with $$ F= \left( {\begin{array}{cc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 ...