Questions tagged [quantum-gate]

For questions regarding usage, performance, implementation, application or theory related to quantum gates.

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Prove $E(R_n(\alpha),R_n(\theta)^n)<\epsilon/3$ from $E\big(R_n(\alpha),R_n(\alpha+\beta)\big)=|1-\exp(i\beta/2)|$

where $E(U,V)=\max_{|\psi\rangle}||(U-V)|\psi\rangle ||=||U-V||$ is the error when $V$ is implemented instead of $U$. See page 196, Quantum Computation and Quantum Information by Nielsen and Chuang. I ...
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Do we need ancillary qubits to implement orthogonal measurements?

Consider an $n$ qubit state $|\psi\rangle$. Let's say I want to implement an $m$ outcome orthogonal measurement on $|\psi\rangle$, where $m \neq n$. Denote the set of $m$ orthogonal measurement ...
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44 views

How is transformation for measurement in an arbitrary basis derived?

I started with Qiskit today and find it very exciting. As a first question I want to understand how to measure an arbitrary state $|\Psi\rangle$ not in the basis of ...
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1answer
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Decomposition of U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate [closed]

Can we express U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate $$ U_1(\lambda) = \begin{pmatrix}1 & 0 \\ 0 & e^{i\lambda}\end{pmatrix}$$ $$ \phi(\delta) = \begin{...
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how to know the appropriate time to know the SWAP gate operation in dipole interaction

Consider dipole-dipole interaction between two qubits,$H_{int} = g \boldsymbol\sigma_{1}\cdot\boldsymbol\sigma_{2}=g(X_1X_2+Y_1Y_2+Z_1Z_2)$. How can I show that by turning on this interaction for an ...
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1answer
76 views

Conditional lower bound on approximate stabilizer rank of magic states

I am currently reading about the approximate stabilizer rank and properties of the same. I will quote the definitions from this paper. The stabilizer rank of a quantum state $|\psi\rangle$ is the ...
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1answer
60 views

Measurement of single qubit operator $U$ which is both Hermitian and unitary with eigenvalues $±1$

Suppose we have a single qubit operator $U$ with eigenvalues $±1$, so that $U$ is both Hermitian and unitary, so it can be regarded both as an observable and a quantum gate. Suppose we wish to measure ...
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2answers
47 views

What is the name of this "ancilla based" process to implement gates

I just want to know if there is a specific name for the implementation of a gate on the top qubit with the help of the bottom qubit, represented on this image: It looks like gate teleportation but it ...
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1answer
39 views

Where does the "correction" in quantum error correction occur, specifically when using repetition codes?

I'm reading the part of the qiskit textbook that deals with this (https://qiskit.org/textbook/ch-quantum-hardware/error-correction-repetition-code.html) and so far it seems as though they're just ...
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124 views

Getting non-Clifford after performing several Clifford gates in qiskit

I'm trying to test Clifford gates in qiskit according to the table in Fault-tolerant SQ, page 101. I tried 4 Cliffords in the test $$-X/2 - X -X/2,Y/2,X/2 - -X/2,Y/2,-X/2$$ using the following code <...
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Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?

p.281 of Nielsen and Chuang's book says that A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, ...
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37 views

Equivalent matrix for operators in qiskit

What is eqivalent matrix to qc.rx(np.pi, 0): ...
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294 views

When can pairs of states be transformed into other pairs of states via unitary mapping?

The states $|+\rangle, |-\rangle$ can be mapped to $|0\rangle, |1\rangle$ by a simple rotation. But if I now have other states ($|\psi_0\rangle, |\psi_1\rangle$) which are not orthogonal, does a ...
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1answer
62 views

Is it possible to perform $z$ rotation in Qiskit with just $x$ and $y$ rotations?

Is it possible to perform $z$ rotation in Qiskit with just $x$ and $y$ rotations? I tried the following: ...
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1answer
121 views

Is the CNOT in the standard three-qubit circuit for the GHZ state necessary?

This is a very basic question about the GHZ state. I know the standard construction: A Hadamard on one qubit, and then CNOT gates with targets on all the other ones. However, why can't I just have $n$...
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41 views

How to apply Euler's formula to $Z$ rotations such as $e^{i\pi/8 Z}$?

I was following the Qiskit textbook and wanted to show that $R_z (\pi/4) = e^{i \pi/8 Z}$. Plugging $\pi/4$ in for $\theta$ in the matrix from this page $ \begin{bmatrix} e^{-i \pi/8} & 0 \\ 0 &...
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1answer
42 views

Decomposition of the multi-controlled gate in tensorflow quantum

In TensorFlow Quantum 0.5.0, the support for Cirq gates that have arbitrary control via the gate.controlled_by function is added. I would like to know which kind of ...
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1answer
53 views

What is "gate length" in quantum computing?

I am working on adding a new provider to Qiskit, and I have to specify the properties of the backend. What is "gate length" (in ns) referring to?
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1answer
81 views

Definition of Deutsch gate and meaning of $\theta$

I'm trying to understand the definition of a Deutsch gate. In particular what does $\theta$ mean in its presentation? Is it derived from the coefficients of the input state or a free parameter or ...
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Writing circuits in Qiskit using only Clifford and T gates

Is there a way in Qiskit to write my circuit using only Clifford and T gates (CX, S, H, T and I think also $S^\dagger$ and $T^\dagger$)? With the function compile (with aer simulator) it gives me some ...
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1answer
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Is the square-root-of-SWAP for a pair of 4-dimensional qudits isomorphic to two square-root-of-SWAPS for two pairs of qubits?

This may be a very naïve question indicative of a lot of confusion, but I am trying to understand more about Hamiltonian simulation. I'm starting to intuit that the $n^{th}$-root-of-SWAP acting on a ...
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1answer
42 views

Applying $R^{'\dagger}_{xz}$ in gate teleportation

In figure 4 of the paper Quantum Teleportation is a Universal Computational Primitive, the authors show a circuit for applying a unitary $U$ fault-tolerantly via teleportation. I'll paste the figure ...
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2answers
81 views

Are composite gates within superconducting hardware implemented as a single pulse or as a series of pulses?

If we have for example a gate $U^{\otimes2}$, then within superconducting hardware, is the $U$ applied onto the first qubit and then the second or is a pulse corresponding to a composite gate (tensor ...
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What are the "higher moments" of the gate fidelity?

Reading the paper Gate fidelity fluctuations and quantum process invariants I came across the concept of higher moments of the gate fidelity, for example in the following excerpt from the introduction:...
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What is the set of generators for the qutrit Clifford group?

According to this article, any Clifford gate, acting on $n$ qubits, can be generated by Hadamard, CNOT, and S gates. What are the set of generators for qutrit Cliffords?
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Why does the output from FANOUT appear on the second and third bits?

Figure 1.16: FANOUT with the Toffoli gate, with the second bit being the input to the FANOUT (and the other two bits standard ancilla states), and the output from the FANOUT appearing on the second ...
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110 views

Qutrit analogues of controlled Z and cc-Z gates

I am trying to look for the qutrit analogues of a controlled-Z, and a cc-Z (Z gate with two controls) for qubits. There is a previous answer that gives a qutrit analogue of a CNOT gate, but does not ...
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1answer
79 views

Cliffordness of the qutrit Hadamard gate

Consider a simple generalization of the Hadamard gate to qutrits, defined as follows. \begin{equation} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ \frac{1}{\sqrt{2}} &...
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438 views

Are these two circuits equivalent in performing controlled time-evolution?

I want to perform the controlled time-evolution of some 2 or 3-qubit Hamiltonian. Say we have this example: $$ H= Z_0\otimes Z_1 + Z_1\otimes Z_2 $$ The circuit performing the time-evolution ...
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1answer
75 views

Can we do error correction for entangled particles?

There are several quantum error correction techniques, such as 3-qubit bit-flip code, and Shor’s 9-qubit code. 3-qubit bit-flip code is a straightforward technique for correcting a single error (...
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Implementation of FANOUT using Toffoli gates

Consider the following implementation of FANOUT using Toffoli gates: I'm confused about the following statement: "the second bit being the input to the FANOUT and the other two bits standard ...
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1answer
51 views

Why isn't FANIN required to be able to simulate all other elements in a classical circuit

In Nielsen and Chuang, there's the following paragraph: The Toffoli gate can be used to simulate NAND gates and can also be used to do FANOUT. With these two operations, it becomes possible to ...
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1answer
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Different statevectors with different quantity of complex numbers for same single-qubit states in Qiskit

I have several questions, but before ask, I want to write some theoretical. As we know, we can represent any single-qubit quantum state by the next representation: $$ |\psi\rangle=c_0|0\rangle+c_1|1\...
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1answer
104 views

How to formulate Dynamical Decoupling passes in Qiskit to improve result upon circuit execution

First, let me say that I am not familiar with the idea of Dynamical Decoupling. The goal of this question is to understand how to set up a circuit with dynamical decoupling to improve my hardware ...
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1answer
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How to use the output state $\frac1{\sqrt2}(|0,f(0)\rangle+|1,f(1)\rangle)$ given by this quantum circuit?

In Nielsen and Chuang, for the following circuit that demonstrates quantum parallelism, I have the following question: since the output state of the circuit is $$\frac1{\sqrt2}(|0,f(0)\rangle+|1,f(1)\...
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How to measure the error rate (or fidelity) of a particular gate (such a H, T) by Qiskit

We know the calibration data of IBM's quantum devices can be retrieved through the randomized benchmarking method, which is included in Qiskit. However, the calibaration data about gates just gives ...
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1answer
110 views

How to design Multi qubit Controlled Z rotations

I need some help in multi-qubit controlled -Z rotation. Below is the qiskit code of triple controlled z rotation ...
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1answer
618 views

Are anti-unitary gates possible?

According to Wigner’s theorem, every symmetry operation must be represented in quantum mechanics by an unitary or an anti-unitary operator. To see this, we can see that given any two states $|\psi\...
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Accuracy of rotational gates on IBM Quantum

As a computer memory is finite, it seems reasonable to assume that a rotational angle of gates $Rx$, $Ry$ and $Rz$ (and $U1$, $U2$ and $U3$ gates as well) has some smallest step which can be ...
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101 views

Does anyone have some references on EigenGate used in Google's cirq framework?

I am trying to understand the code below but can not find any references other than the code itself. https://github.com/quantumlib/Cirq/blob/v0.11.1/cirq-core/cirq/ops/eigen_gate.py I am not clear ...
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2answers
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Can we use reversible computation to construct oracle circuits?

One of the question while I discussed with my colleague in the math department was the construction of oracle circuit. In computer science, specifically in algorithm, we take oracle as granted and ...
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1answer
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About projective measurement and eigenspace

I have read an article Unified derivations of measurement-based schemes for quantum computation. In page 3, Section A it says that The classical outcome j corresponds to the measurement of $(U^\dagger ...
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112 views

Can someone explain using H and T gates repeatedly?

The Qiskit textbook says it is used because $R_x$, $R_y$, and $R_z$ are not accurate single qubit rotations. Can someone elaborate what the repeated $H$ and $T$ gates actually do? In what scenario ...
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The use of modulo 2 in state representation after CNOT

The following circuit with a CNOT gate has the following effect on a computational basis state $|a, b\rangle$, where all additions are done modulo 2. Why is the state of the second qubit changed to $...
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Defining native gate dictionary in pyGSTi

In pyGSTi in order to construct Randomized Benchmarking circuits, we first need to define a pspec object that contains information about the number of qubits, basis ...
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1answer
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How to derive the rotations caused by the H gate?

In Nielsen and Chuang, there's the following paragraph: The Hadamard operation is just a rotation of the sphere about the ˆy axis by 90◦, followed by a rotation about the ˆx axis by 180◦. I am ...
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1answer
223 views

Why is H gate called a ‘square-root of NOT’ gate?

In Nielsen and Chuang, there's the following paragraph: I understand that \begin{align*} \sqrt{NOT} = \frac{1}{2}\left( {\begin{array}{*{20}{c}} \sqrt 2 e^{i\pi / 4}&\sqrt 2 e^{-i\pi / 4}\\ \sqrt ...
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3answers
742 views

Why do quantum gates have to be unitary?

In my textbook, it's said the unitarity constraint is the only constraint on quantum gates. Any unitary matrix specifies a valid quantum gate! Why do quantum gates have to have to be unitary? How do ...
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1answer
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Measuring tensor products of Pauli operators

Is there a neat way to derive and efficiently implement a measurement circuit for tensor products of arbitrary Pauli operators like $XZZXZ$ in Qiskit ? I tried using the ...
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Apply a custom gate to qubits in separate quantum registers

Let there be a custom gate (simply called gate that takes $N$ qubits as input. Also, let $A$ of those qubits be in first_register...

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