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 ...
3
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1answer
56 views
How to obtain Y rotation with only X and Z rotations gates?
Let's say you have a system with which you can perform arbitrary rotations around the X and Z axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the Y axis?
...
5
votes
1answer
42 views
Mapping Algebraic Normal Form of Exclusive Sum of Products to Toffoli Network
How does one map an ANF to a toffoli network? Is there a straight-forward procedure for doing this?
For example, given the ANF for the Sum function of an adder:
$$S = A \oplus B \oplus C \oplus ABC$$
...
5
votes
2answers
68 views
Clarification needed: “Simulation” of $e^{-iHt}$ and its time complexity
On page 3 here it is mentioned that:
However, building on prior works [32, 36, 38] recently it has been
shown in [39] that to simulate $e^{−iHt}$ for an $s$-sparse Hamiltonian
requires only $\...
1
vote
1answer
76 views
Applying Group Leaders Optimization to Quantum Belief Systems
Context: I am particularly interested in quantum cognition & would like to use a tool like pyZX to perform the following types of optimizations.
In Preparing a (quantum) belief system they "...
1
vote
2answers
32 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 $A$
$$ \frac{1}{4} \left[\begin{matrix}
15 & 9 & 5 & -3 \\
9 & 15 & 3 &...
4
votes
1answer
77 views
Understanding the Group Leaders Optimization Algorithm
Context:
I have been trying to understand the genetic algorithm discussed in the paper Decomposition of unitary matrices for finding quantum circuits: Application to molecular Hamiltonians (Daskin &...
2
votes
1answer
56 views
What quantum gate is XNOR equivalent to?
The standard way to implement a reversible XOR gate is by means of a controlled-NOT gate or CNOT; this is the "standard quantum XOR operation". Physics.Stackexchange
Is there a "standard quantum XNOR ...
5
votes
1answer
63 views
How can I decompose a gate into $\{\mathrm{CNOT}, \mathrm{H}, \mathrm{P}(\theta)\}$?
I am working with the set $\{\mathrm{CNOT}, \mathrm{H}, \mathrm{P}(\theta)\}$
where $\mathrm{H}$ is the Hadamard gate, and $\mathrm{P}(\theta)$ is the phase gate with angle $\theta$.
I want to build ...
5
votes
1answer
144 views
Decomposition of an arbitrary 1-qubit gate into a specific gateset
Any 1-qubit special gate can be decomposed into a sequence of rotation gates ($R_z$, $R_y$ and $R_z$). This allows us to have the general 1-qubit special gate in matrix form:
$$
U\left(\theta,\phi,\...
6
votes
1answer
171 views
Decomposition of arbitrary 2 qubit operator
As you know, universal quantum computing is the ability to construct a circuit from a finite set of operations that can approximate to arbitrary accuracy any unitary operation.
There also exist some ...
13
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0answers
165 views
Explicit Conversion Between Universal Gate Sets
I'm interested in the conversion between different sets of universal gates. For example, it is known that each of the following sets is universal for quantum computation:
$\{T,H,\textrm{cNOT}\}$
$\{H,...
9
votes
2answers
256 views
Automatic compilation of quantum circuits
A recent question here asked how to compile the 4-qubit gate CCCZ (controlled-controlled-controlled-Z) into simple 1-qubit and 2-qubit gates, and the only answer given so far requires 63 gates!
The ...
8
votes
6answers
392 views
How to construct a multi-qubit controlled-Z from elementary gates?
For the implementation of a certain quantum algorithm, I need to construct a multi-qubit (in this case, a three-qubit) controlled-Z gate from a set of elementary gates, as shown in the figure below.
....
5
votes
1answer
77 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-...
3
votes
1answer
76 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 ...
6
votes
1answer
157 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
votes
1answer
100 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 ...
3
votes
2answers
150 views
Components for realizing 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 components are ...
5
votes
2answers
392 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 ...
9
votes
1answer
509 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 ...
6
votes
1answer
204 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
votes
1answer
142 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 ...
10
votes
2answers
233 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 ...
7
votes
2answers
419 views
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 ...
8
votes
2answers
130 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}...
9
votes
1answer
200 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?
8
votes
2answers
245 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 ...
10
votes
1answer
230 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 ...
11
votes
2answers
357 views
Circuit of controlled-Unitary gate
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
votes
2answers
258 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 ...
10
votes
1answer
167 views
How are quantum gates realised, in terms of the dynamic?
When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions.
In some sense, these are rather mysterious objects, in that they perform "...
12
votes
1answer
94 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 ...
