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

$\newcommand{\bra}{\left<#1\right|}\newcommand{\ket}{\left|#1\right>}\newcommand{\bk}{\left<#1\middle|#2\right>}\newcommand{\bke}{\left<#1\middle|#2\middle|#3\right>}$ I'...
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### 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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### 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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### 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 $X$ ...
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### 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 ...
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### 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 ...
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### 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_{...
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### 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 ...
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### Nielsen and Chuang's proof for 'approximating arbitrary unitary gates is generically hard'

The following statement is found on the page 199 of Nielsen and Chuang's book (10th Anniversary Edition) in the proof for the fact that 'approximating arbitrary unitary gates is generically hard': ...
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### Transferring classical OR gate in a quantum gate

I would be interested to know how to transform the classic OR gate into a quantum gate. I thought a little about myself. The OR gate can also be rewritten as a NAND gate: So, I have now tried to ...
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### Minimal quantum OR circuit

The quantum OR circuit between $|a\rangle$ and $|b\rangle$ can be made out of 1 Toffoli and 2 CNOT gates, 1 ancillary qubit. Is there any other implementation? Or is this the minimal in the sense of ...
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### Making a maximally mixed 2-qubit state in the IBM Q

I am trying to make a 2-qubit maximally mixed state $\mathbb{I}/4$ where $\mathbb{I}$ is the identity $4\times 4$ matrix. I know that, for a maximally mixed 1-qubit state I can use a Hadamard gate, ...
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### How is it possible to implement unitary operator when its size is exponential in inputs?

A quantum circuit can use any unitary operator. Its matrix is exponential in the number of input bits. In practice how can this ever be possible (aside from operators which are tensor products), i.e. ...
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### Implementing gate with two parameters using Qiskit in Python

I am trying to implement the HHL algorithm (for solving $Ax=b$). I am assuming $A$ to be unitary and Hermitian so that I can find the Hamiltonian simulation for it easily. For any $A$ to be Hermitian ...
I am trying to implement VQE in pyQuil and am dumbfounded by how to measure the expectation value of a general Hamiltonian on $\mathbb{C}^{2^n}$ i.e. determine $\langle\psi , H \psi\rangle$ on a ...