Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

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.

212 questions
7
votes
1 answer

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: …
Ntwali B.
  • 443
  • 2
  • 9
3
votes
1 answer

Gate sequence for exponential of product of Pauli Z operators

I want to compile $$\exp(-i \theta \sigma_i^z \sigma_j^z)$$ down to a gate sequence of single qubit rotations and CNOTs. How do I do this? What is the general procedure for compiling a unitary $U$ to an elementary gate sequence? (As I understand it,…
Joe
  • 71
  • 6
3
votes
1 answer

What is the function group generated by generalised Toffoli gates?

I am trying to define a mathematical framework that starts from a generic function $f$, which I can synthesis as a circuit of generalised Toffoli gates -- i.e. $m$ controls, 1 target. I would like to read of a framework similar to the one used for…
Daniele Cuomo
  • 1,742
  • 8
  • 21
3
votes
0 answers

Brute force gate decomposition of (specific) 4 qubit unitary matrix

I have a specific 4-qubit 16x16 unitary matrix $U$ with $9$ parameters. My goal is to find a gate decomposition in terms of e.g. {Rx, Ry, Rz, CNOT}. I feel like there must be a brute force way to find that, but the parametrization is making it…
Korbinian
  • 145
  • 5
3
votes
1 answer

Tool to verify $CNOT$ (or any interacting 2-qubit gate)

Is there any tool to define a circuit and verify if it works as desired? It would be interesting to find ways of performing interacting gates - e.g. CNOT gate - between non adjacent qubits. Hence I'd like an efficient way to define a circuit with…
Daniele Cuomo
  • 1,742
  • 8
  • 21
3
votes
0 answers

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 and the CNOT. The unitaries $U_i$ act on an…
Georg
  • 131
  • 3
2
votes
0 answers

Creating a unitary for binary encoding with respect to already encoded index states

Let us say that there are two quantum registers qr1 and qr2. Now the qr1 is in the state $\sum_i |x_i\rangle$(here $x_i$ is binary encoded value upto some precision) and originally qr2 is $|0\rangle$, the net state is, $$|\psi_0\rangle=\sum_i…
Parmeet Singh EP 066
  • 323
  • 1
  • 8
2
votes
1 answer

Is it possible to decompose $\land(UXU^\dagger)$ in one-qubit operations and only a single $\land(X)$?

Let $U,V$ being any unitary. Is it possible to decompose $\land(UXU^\dagger)$ in one-qubit operations and only a single $\land(X)$? Something like the following: $\land(UXU^\dagger) \equiv (\mathbb{I}\otimes V)\land(X)(\mathbb{I}\otimes…
Daniele Cuomo
  • 1,742
  • 8
  • 21
1
vote
0 answers

Proving that linear reversible boolean functions are permutation functions

The answer to this question begins with a statement of my interest: |⟩=|()⟩ where :2→2 is a reversible Boolean function. These are exactly the permutations of bitstrings Assuming one has to perform $U$ followed by measuring the system. If $f$ is…
Daniele Cuomo
  • 1,742
  • 8
  • 21
1
vote
2 answers

Decomposition of $\exp(-i (X_1X_2 + Y_1Y_2) X_3)$

The three-body terms $\exp[-i\theta(X_1X_2+Y_1Y_2)X_3]$ and $\exp[-i\theta(X_1X_2+Y_1Y_2)Y_3]$ lead to unitaries of the form $$ \begin{bmatrix} 1 & & & & & & & \\ & 1 & & & & & & \\ & & a & & & b & & \\ & & & a & b & & & \\ & & & b & a & & &…
NaturalLog
  • 159
  • 7
0
votes
1 answer

Can we express $CX_{2,1}CX_{1,2}$ as single standard 2-qubits gate?

I'd like to know if the above circuit can be synthesised as any single standard 2-qubit gate -- e.g. an Ising gate. Eventually, other 1-qubit correcting gates. EDIT: with standard I mean any gate that has practical implication. Especially when this…
Daniele Cuomo
  • 1,742
  • 8
  • 21
0
votes
1 answer

When are the following equivalences correct?

I can't figure out how the equivalences in the picture hold. The picture comes from this recent publication on PRA. EDIT: I think I might have been mislead by the gate represenation. In fact, the gate is more common to express the $CZ$ gate and not…
Daniele Cuomo
  • 1,742
  • 8
  • 21