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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8
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2answers
340 views

Number of gates required to approximate arbitrary unitaries

If I understand correctly, there must exist unitary operations that can be approximated to a distance $\epsilon$ only by an exponential number of quantum gates and no less. However, by the Solovay-...
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2answers
250 views

Arbitrary powers of NOT and SWAP

The square-root of not and square-root of swap gates are often singled out for discussion of gates displaying important properties relating to quantum computers. How do I define arbitrary (non-...
8
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2answers
362 views

Is it possible to realize CNOT gate in 3 dimension?

CNOT gates have been realized for states living in 2-dimensional spaces (qubits). What about higher-dimensional (qudit) states? Can CNOT gates be defined in such case? In particular, is this possible ...
3
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1answer
118 views

Doing maths with controlled-half NOTs

In Quantum Computation with the simplest maths possible there is a section titled "Doing maths with a controlled-half NOT" which covers a reversible-(N)AND circuit with controlled-half NOTs. What ...
9
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1answer
353 views

Basic approximation in Solovay-Kitaev algorithm

I read the Solovay-Kitaev algorithm for approximation of arbitrary single-qubit unitaries. However, while implementing the algorithm, I got stuck with the basic approximation of depth 0 of the ...
4
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1answer
310 views

Square root of NOT as a time-dependent unitary matrix

I want to express the square root of NOT as a time-dependent unitary matrix such that each $n$ units of time, the square root of NOT is produced. More precisely, I want to find a $U(t_0,t_1)$ such ...
10
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1answer
1k views

How to implement a matrix exponential in a quantum circuit?

Maybe it is a naive question, but I cannot figure out how to actually exponentiate a matrix in a quantum circuit. Assuming to have a generic square matrix A, if I want to obtain its exponential, $e^{A}...
5
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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 ...
8
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2answers
254 views

Construct Controlled-$G^{\dagger}$ from known Controlled-$G$

Let there be a known a scheme (quantum circuit) of Controlled-G, where unitary gate G has G$^†$ such that G≠G$^†$ and GG$^†$=I (for example S and S$^†$, T and T$^†$, V and V$^†$, but not Pauli and H ...
7
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2answers
2k views

Expressing “Square root of Swap” gate in terms of CNOT

How could a $\sqrt{SWAP}$ circuit be expressed in terms of CNOT gates & single qubit rotations? CNOT & $\sqrt{SWAP}$ Gates Any quantum circuit can be simulated to an arbitrary degree of ...
10
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1answer
2k views

Implementing a CCCNOT gate using only Toffoli gates

A CCCNOT gate is a four-bit reversible gate that flips its fourth bit if and only if the first three bits are all in the state $1$. How would I implement a CCCNOT gate using Toffoli gates? Assume ...
7
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1answer
383 views

Quantum XOR Linked List Construction

After getting help here with XNOR & RCA gates I decided to dive into XOR Swaps & XOR linked lists. I was able to find this explanation for quantum XOR Swapping which seems sufficient for the ...
6
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1answer
373 views

Quantum Ripple Carry Adder Construction

There is an excellent answer to How do I add 1+1 using a quantum computer? that shows constructions of the half and full adders. In the answer, there is a source for the QRCA. I have also looked at ...
12
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2answers
495 views

Quantum XNOR Gate Construction

Tried asking here first, since a similar question had been asked on that site. Seems more relevant for this site however. It is my current understanding that a quantum XOR gate is the CNOT gate. Is ...
9
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3answers
2k views

How to implement the “Square root of Swap gate” on the IBM Q (composer)?

I would like to simulate a quantum algorithm where one of the steps is "Square root of Swap gate" between 2 qubits. How can I implement this step using the IBM composer?
10
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2answers
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Implementation of the oracle of Grover's algorithm on IBM Q using three qubits

I am trying to get used to IBM Q by implementing three qubits Grover's algorithm but having difficulty to implement the oracle. Could you show how to do that or suggest some good resources to get ...
7
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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 ...
11
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1answer
2k views

How can a controlled-Ry be made from CNOTs and rotations?

I want to be able to applied controlled versions of the $R_y$ gate (rotation around the Y axis) for real devices on the IBM Q Experience. Can this be done? If so, how?
10
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2answers
573 views

Shortest sequence of universal quantum gates that correspond to a given unitary

Question: Given a unitary matrix acting on $n$ qubits, can we find the shortest sequence of Clifford + T gates that correspond to that unitary? For background on the question, two important ...
15
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1answer
720 views

Obtaining gate $e^{-i\Delta t Z}$ from elementary gates

I am currently reading "Quantum Computation and Quantum Information" by Nielsen and Chuang. In the section about Quantum Simulation, they give an illustrative example (section 4.7.3), which I don't ...
14
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2answers
2k views

Given a decomposition for a unitary $U$, how do you decompose the corresponding controlled unitary gate $C(U)$?

Suppose we have a circuit decomposition of a unitary $U$ using some universal gate set (for example CNOT-gates and single qubit unitaries). Is there a direct way to write down the circuit of the ...
13
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2answers
652 views

What is the mathematical justification for the “universality” of the universal set of quantum gates (CNOT, H, Z, X and π/8)?

In this answer I mentioned that the CNOT, H, X, Z and $\pi/8$ gates form a universal set of gates, which given in sufficient number of gates can get arbitrarily close to replicating any unitary ...
14
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1answer
243 views

How does approximating gates via universal gates scale with the length of the computation?

I understand that there is a constructive proof that arbitrary gates can be approximated by a finite universal gate set, which is the Solovay–Kitaev Theorem. However, the approximation introduces an ...

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