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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Consecutive phased X rotation gates simplification
I have two consecutive phased X rotation (see cirq PhasedXPowGate gate definition), how to find both t' and p' angles (according to previoux cirq definition) so that two consecutive PhasedXPowGate are ...
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0
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CNOT circuit synthesis with Gauss elimination. Explanation and beyond?
This paper introduces to the synthesis of a (optimal) circuit of CNOTs only; starting from a parity map encoded into a matrix.
It is based on Gaussian Elimination.
This is an important result, which ...
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0
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Can the optimal synthesis of a parity matrix generate complex combinations of CXs?
Consider the method described in Optimal Synthesis of Linear Reversible Circuits.
I am led to think that, for any parity matrix given in input to such a method, the output would be a sequence of CX ...
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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 ...
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1
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Math Behind $X$ Gate With Arbitrary Phase is equivalent to $ZXZ$ Gate
An X gate where there is a phase shift $\phi$ to the applied sinusoidal wave $U = e^{-i\frac{\theta}{2}(cos(\phi)\sigma_x+sin(\phi)\sigma_y)}$ is equivalent to a series of gates $Z_{-\phi}X_{\theta}Z_{...
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0
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Application of transformation $U_d$ that maps any qudit state to $|d-1\rangle$
When giving examples of universal gate sets in the paper Qudits and High-Dimensional Quantum Computing, the authors first define the transformation that maps any given qudit state to $|d-1\rangle$:
$$
...
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2
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1
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Constructing a controlled phase gate from given gates
As part of a project in a quantum computing course we were asked to classically simulate the quantum phase estimation algorithm, which has inverse QFT as one of its components.
On the Wikipedia page ...
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Global (Ising) Gates and ZX-calculus representation
I could find from this source -- but also from other works on ZX-calculus -- the following extract:
This looks to me as a generalisation of a 2-qubit Ising gate to an $n$-qubit global Ising gate. ...
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How to find the canonical form (i.e., phase-free representation) of a unitary matrix?
While reading Weiden and others' recent paper: Improving Quantum Circuit Synthesis with Machine Learning, I came across the notion of canonically representing a unitary matrix. More precisely, two ...
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1
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Exact synthesis of Toffoli gate from CNOT and rational single-qubit gates?
Is it possible to implement a Toffoli gate exactly using just CNOT gates and single qubit complex rational gates (i.e. with entries in $\mathbb{Q}(i)$), possibly with ancillas?
I know this works with ...
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2
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Gate synthesis with parametrised precision
I am wondering whether Qiskit (or other quantum program language) can perform gate synthesis with parametrised precision.
I tried with
...
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How can I simulate the following 2×2 Hamiltonian $e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}}$?
How can I simulate the following 2×2 Hamiltonian $$ e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}}|\Psi\rangle$$
ie. how to rewrite that matrix exponential in terms of other, well-used ...
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2
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Is there a matrix exponential $e^{iA}$ gate in IBM Quantum Experience?
Is there a gate that can perform the matrix exponential operation
$$e^{iA}|\Psi\rangle$$
in IBM quantum experience API?
What is the name and symbol for this type of gate (or some other gates that can ...
- 455
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votes
1
answer
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Computing the Bloch sphere representation of an arbitrary operator in $U(2)$
Computing the Matsumoto-Amano normal form of an operator in $U(2)$ involves finding the Bloch sphere representation of said operator, see Remarks on Matsumoto and Amano’s normal form for single-qubit ...
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1
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How many two-qubit controlled gates do you need to simulate any CU gates where U is a diagonal matrix?
Assuming we have n qubit, the first qubit is a control qubit, and the rest are the targets of $U$.
If $U$ is a diagonal matrix, is there any theory to find the minimum number of two-qubit controlled ...
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0
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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. ...
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1
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How to convert a basic matrix into a quantum circuit?
Classical gates are not invertible, but larger expressions made from those gates can be invertible. One example of an invertible function is the function $f(A,B,C) = X,Y,Z$:
$X = A \ B \ | \ \neg B \ ...
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1
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Converting a Matrix to a Gate in OpenQasm 2
I am a beginner when it comes to quantum computing so forgive me if this is a dumb question. Does anyone know how to create a gate from any matrix on OpenQasm2? Specifically, can anyone provide any ...
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Universality with Toffoli + Hadamard
If I take the two gates Hadamard and Toffoli, then it is clear that I cannot simulate an arbitrary $n$ qubit unitary on $n$ qubits because both matrices are real, so there's no access to the complex ...
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perform a SWAP measurement using local operations and classical feedback
I am interested in performing a SWAP measurement, namely, measure 2 qubits and project their state onto either the triplet state manifold $\{|00\rangle, |11\rangle, |01\rangle + |10\rangle\}$ or the ...
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2
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Can a $CX_{1,2}\cdot CX_{2,1}$ be synthesised to some $CU$ plus local gates?
Can the above circuit be synthesised to an operation where there is only one control qubit? I.e. a controlled-unitary gate, eventually surrounded by local gates.
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Number of distinct permutation classes up to multiplication by Clifford elements
Question
The number of permutation gates on $n$ qubits is $2^n!$. Define an equivalence relation on these gates by $p_1 \approx p_2$ iff $p_1 = C_L p_2 C_R$ where $p_1, p_2$ are $n$-qubit permutation ...
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1
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Unitary to circuit in qiskit
I have a program that determine the unitary matrix of a unknown gate in a quantum circuit and then it checks in the standard gate list to get the name of unknown gate. It is guaranteed that unknown ...
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0
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How to Trotterize a CNOT gate?
I came across a paper that said that they Trotterized a CNOT gate into 4 blocks of CU gates where the CU gate parameters are specified. This was all done on Qiskit. How does this Trotterization ...
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2
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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 ...
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1
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How to construct solution based on the Schrödinger equation and split it into gates?
To the best of my knowledge, the gate notation forms the quantum programming. For instance, I use qiskit, pennylane, etc. products to see how the algorithms do their job. At the same time the "...
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1
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How to implement the cross term (multi-qubit) in the square of the finite difference operator?
I am trying to simulate the Hamiltonian evolution of the 1+1D $\lambda\phi^4$ scalar field theory by digitising it and encoding on a quantum computer. The process of digitising is taken from this ...
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0
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Non-local $CNOT$ By means of Ising gates
Consider the circuit below.
This is almost the same as the standard protocol to perform a non-local $CNOT_{0,3}$. The only difference is that I decomposed the upper local $CNOT_{0,1}$ into one Ising ...
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Transforming an unkown phase into unkown bit values on bell states
Consider the state $|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)$.
The $|\Psi^\pm\rangle$ state is a bell state up to an unkown phase.
I am looking for a sequence of single-qubit ...
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What is the minimum number of non-Clifford gates does it take to prepare a superposition over all "two-hot" basis vectors?
The generalized W state:
$$W_n=\frac{1}{\sqrt{n}}(|100\cdots 0\rangle + |010\cdots 0\rangle + \ldots + |00\cdots 01\rangle)$$
is often thought of as the uniform superposition over all "one-hot&...
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How to construct common classical gates with CNOT circuit?
How can I construct AND, OR, NAND, NOR with CNOT gates.
First off, this other question describes how to make them with matrices.
Theoretically I can construct all those gates already. I know how to ...
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4
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1
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Shortest depth on Clifford+T to decompose a Toffoli
I am looking for a reference providing a circuit that has the smallest possible depth, without ancilla, once the Toffoli has been decomposed on Clifford+T gateset, where Clifford is generate by cNOT, ...
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1
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Implementing controlled rotation for FRQI by using controlled Ry and NOT Gate
I was reading about Flexible Representation of Quantum Images (FRQI) encoding in Qiskit textbook. It says that, given $$\{\theta_0, \theta_1, ..., \theta_{4^{n}-1}\} \quad (\theta_i \in [0,\pi/2])$$ ...
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1
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qiskit : can you get circuit from unitary matrix?
In qiskit you can get a unitary matrix from a circuit (circuit to unitary matrix example). Is the opposite direction possible? Can you input a unitary matrix and have qiskit come up with a circuit? If ...
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What's the most efficient decomposition in terms of T-count of the 4-qubit Toffoli with 1 ancilla?
When decomposing the 4-qubit Toffoli in the Clifford+T universal gate set with 1 ancilla qubit, what is the most efficient implementation one can get in terms of T-count? I can only find papers that ...
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1
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How to synthesize function $f(x)$ in amplitude encoding
In computational basis encoding, the way to encode $f(x)$ is known - a classical circuit is converted to a quantum circuit which takes $|x\rangle|0\rangle \to |x\rangle|f(x)\rangle $. I wonder how I ...
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Implementing Odd Permutations Without Ancilla Bit
The paper says that
The inversion $\alpha \mapsto \alpha^{-1} $ (where 0 is mapped to 0)
can be seen as a permutation on $\mathbb F_{256}$. This permutation is odd, while
quantum circuits with NOT, ...
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3
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1
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Cirq : Reference for Toffoli decomposition
I was trying to find a reference for the 7 T-gate decomposition of the Toffoli gate given by Cirq. The decomposition originates from the the one used for CCZPowGate as given in the doc string here
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0
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PyZX optimisation steps for Clifford circuits
Given the following ZX-diagram
It should represent some random Clifford circuit (LC means Local Clifford).
As far as I got, any Clifford circuit can be transformed into a ZX-diagram like the above, i....
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3
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How to decompose a multi qubit Clifford unitary into a sequence of clifford gates
What are the algorithms that allow to decompose any given multi qubit Clifford unitary into elementary Clifford operations (e.g. Pauli+CNOT, with no T gate)?
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Faithful description of a photonic setting with the circuit model
The above picture comes from this paper.
The circuit on the left and the one on the right are equivalent (up to the basis).
However, there is an important difference: the circuit makes the input -- i....
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2
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Does phase kickback require the system to be in the eigenstate?
I've been watching this video for the introduction to phase kickback. And here's a diagram:
I got confused if we really need $|\psi_k\rangle$ to be an eigenstate to make the kickback work. It seems ...
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1
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Best implementation for logical CNOT on Shor's code?
As the Shor's code is a CSS code, it admits a transversal implementation of logical CNOT.
An immediate implementation may perform 9 (reversed) CNOT, by respecting the order of the qubits.
However. ...
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How to initialize a state of the form$\frac{1}{\sqrt{2}}(|\texttt{++}\rangle + |\texttt{--}\rangle$) in the circuit model?
I wonder how to initialise a Bell-like state, in the circuit model, where instead of standard $|\Phi^{\texttt{+}}\rangle$, the entanglement is in the x-basis.
Hence a state $\frac{1}{\sqrt{2}}(|\...
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2
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On the photonic implementation of Shor's code
The above picture comes from this paper.
I can see that the standard Shor's code has been re-designed.
I have two main doubts:
I can't figure out in figure (b) how the setting inputs the state $\...
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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 ...
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Is there a name for a gate that 'moves' one qubit to a new position via multiple SWAP gates?
Let's say there is a qubit at position $i$, and I want to move it to position $i'$. Without loss of generality, let's say $i < i'$. By 'move it' I mean, perform multiple $SWAP$ operations so that ...
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2
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CNOT chain vs CNOT fountain in qiskit
I was going through qiskit's synthesis module, their methods take an argument called cx_structure which has two possible values, ...
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0
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37
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Techniques to parallelize controlled-unitaries controlled by the same qubit but acting on different target qubits
I need to find a way to parallelize a set of controlled-unitaries that are all controlled by the same qubit and are targetting $n$ different qubits. The main constraint that I have is that I can only ...
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What is the correct name of this quantum gate? Possibly state control gate
Let $\vec v \in \mathbb{C}^2 $ be the following quantum state:
$$
\vec v = \frac{1}{\sqrt{2}}\begin{bmatrix}
v_{1} \\
v_{2} \\
\end{bmatrix},\space \lvert v_1 \rvert = 1,...
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