# 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.

123 questions
Filter by
Sorted by
Tagged with
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 ...
388 views

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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
101 views

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 ...
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 ...
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 ...
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....
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 ...
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 ...
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'...
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?
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 ...
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 $X$ ...
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 ...
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 ...
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 \mathbf{W}_j=\begin{pmatrix} |\alpha|2\cos{(\phi_{...
### 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 ...
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 ...