forky40
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How do you represent a Hadamard gate as a product of $R_x$ and $R_y$ gates?
8 votes

If you're not concerned with global phase then the following works using only two rotation gates: \begin{align} R_y\left(-\frac{\pi}{2}\right) R_x\left(\pi\right) &= \exp \left(i\frac{\pi}{4}Y\...

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Does ${\rm tr}(\Pi \rho) = 1$ imply $\Pi\rho\Pi=\rho$?
Accepted answer
6 votes

Yes its true. Define another orthogonal projector $\Pi_\perp$ such that $\Pi + \Pi_\perp = I$ and write $\rho$ in terms of a spectral decomposition \begin{equation} \rho = \sum_k \lambda_k(\rho) |\...

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

The two pages you link are using opposite conventions, the first defines \begin{align} R_z(\theta) &:= \exp(i \theta Z /2)\\ &= \cos (\theta/2) + i \sin (\theta/2)Z\\ &= \begin{pmatrix} \...

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Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?
Accepted answer
6 votes

No such (orthonormal) basis can exist. An orthonormal basis $\{|\psi_i\rangle\}$ requires $\langle \psi_i | \psi_j \rangle = 0$ for $i\neq j$, and so clearly \begin{align} [\rho_i, \rho_j] &= |\...

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What is the square root of the NOT gate?
6 votes

As mentioned already, both of those unitaries are the same up to a global phase. It might be useful to think about how you can actually arrive at one of these definitions in terms of the "Not ...

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Is a "kernel" just the quantum equivalent of classical SVMs?
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6 votes

Consider a simple implementation of a Support Vector Machine (SVM) that finds a hyperplane (defined by its normal vector $w$) that maximally separates vectors $\{v_1, \dots, v_m\}$ according to their ...

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How does the NOT gate generalize beyond binary?
6 votes

As the previous answer mentions, how a controlled qudit gate is defined is up to a choice of convention. This paper contains a few examples of intuitively appealing definitions for controlled qudit ...

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What are magic states?
6 votes

In addition to the accepted answer and @user1271772's examples, here is a circuit primitive referred to explicitly as a "T-gate gadget" in [1] (originally appearing in [2]): where ...

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What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?
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5 votes

Summary: There is a solution for expressing a tridiagonal matrix of the form you've provided for arbitrary $n$ in terms of Pauli operators using recursion. This procedure is given at the bottom of ...

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How to write the three-qubit GHZ state in the Pauli basis?
5 votes

Summary: The expression you're looking for is: $$ \frac{1}{4} \left[ (III + IZZ + ZIZ + ZZI) + (XXX - XYY - YXY - YYX)\right] $$ where Pauli string notation like $XYX$ denotes $\sigma_1 \otimes \...

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Does QFT exploit entanglement?
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5 votes

Yes, the formula you have shows that applying QFT to a given computational basis state $|j\rangle = |j_1 j_2 \dots j_n\rangle$ results in an unentangled output state. However when applied to ...

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What does it mean that 1 qubit can do the job of 1 ebit (entanglement bit)? (second Bennett's law)
Accepted answer
5 votes

Without sacrificing any generality we can define an ebit as a Bell state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ shared between two parties $A$ and $B$, and since we're concerned with ...

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Finding the norm of a Hamiltonian
5 votes

The spectral norm $\|H\|$ (sometimes denoted$^1$ $\|H\|_2$ or $\|H\|_\infty$) in this case is the largest eigenvalue of $H$. There's no meaningful bound for this number without having additional ...

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SWAP Test as a Projective Measurement
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5 votes

There are two different ancillas floating around, one used in $|\psi\rangle$ and another to conduct the swap test later on: In the above picture with subsystems explicitly labeled we have \begin{...

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How can measuring a particle in a GHZ state leave behind a maximally entangled pair?
Accepted answer
5 votes

As mentioned in the article, you can rewrite the GHZ state as \begin{align} \frac{1}{\sqrt{2}} (|000\rangle + |111)&= \frac{1}{2\sqrt{2}}(|000\rangle + |111 \rangle + \overbrace{|001\rangle + |...

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Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?
Accepted answer
5 votes

That quantity appears to be identical to Holevo information, which turns out to be the upper bound on how much classical information you can transmit using a quantum channel [1]. More generally the ...

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What is the difference between transmon and Xmon qubits?
5 votes

In one sense, the Xmon qubit is a transmon qubit, in that they both operate in the $E_J>>E_c$ regime of the CPB Hamiltonian and take advantage of the exponentially suppressed charge noise vs. ...

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Quantum teleportation of a mixed state through a pure state?
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4 votes

The teleportation should behave just the same with a mixed state as it does with a pure state. I'm going to assume a bit of familiarity with how teleportation works for pure states, as you can find ...

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Python shorthand for tensor product (Kronecker product)
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4 votes

Python currently doesn't support an operator for Kronecker products. Note how the @ symbol works: when you write the statement A @ B, Python$^1$ checks the objects A and B for a __matmul__ method and ...

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Biggest variance of $h=\sum_i H_i$?
Accepted answer
4 votes

Here is an approach that requires no specific knowledge about $|\psi\rangle$ whatsoever. In your description you implied that each $H_i$ has the same maximum and minimum eigenvalues $\lambda_m$ and $\...

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How is quantum machine learning reversible?
Accepted answer
4 votes

There are a couple of ways reversibility might be coming into play in this context. The first is that the measurement at the end of the circuit will be typically be an irreversible step. For example ...

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What does the quantum part of the quantum support vector machine actually do?
Accepted answer
4 votes

The basic idea of how the quantum feature map works is that you're using a quantum computer to map each input datapoint $x$ from your training domain $\mathcal{X}$ into a quantum state $|\phi(x)\...

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How is a qubit in superposition between $|0\rangle$ and $|1\rangle$ different from a mixture of $|0\rangle$ and $|1\rangle$?
Accepted answer
4 votes

The mixed state is invariant under unitary operations; no choice of $U$ that you might apply to $\rho_2$ will change its output statistics from 50/50 distribution of "0" versus "1"....

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Is the square root of SWAP gate "maximally entangling"?
Accepted answer
4 votes

It seems like the article you're referencing is defining "maximally entangling" as "capable of producing Bell states from product states". However there are other ways to describe ...

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Can we conclude that errors on Sycamore are Poisson-distributed Pauli errors?
Accepted answer
4 votes

The model's accuracy is purely empirical observation. The error trend (Fig 4, or 41:50 in the video) demonstrates that the error of the system (cross entropy fidelity with respect to simulated results)...

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Looking for papers that are pessimistic about the feasibility of a quantum computer
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4 votes

You can look up work by Gil Kalai, who is a longstanding and outspoken critic of quantum computing (his most recent essay: Kalai, 2019). He often bases his view on assumptions that I entirely disagree ...

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Why doesn't the Gottesman-Knill theorem render quantum computing almost useless?
4 votes

Another way to think about this: To simulate what goes on in a quantum computer we have to do a lot of matrix math using $(2^N \times 2^N)$ matrices$^1$, and the action of (most) of the clifford gates ...

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For which quantum algorithms is it sufficient to know $\langle \sigma_1...\sigma_{N} \rangle$ and/or $\langle \sigma_i \rangle$?
3 votes

This is not a complete answer but describes a case where knowing polynomially many Pauli expectation values is not sufficient to solve the same problem. Consider that the set of expectation values for ...

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Bias in the results of superposition measurements on IBMQ Backends, qiskit
3 votes

Asymmetric readout error While its hard to make strong claims about the noise characteristics of any given quantum device, one explanation for what you're observing is readout error. For ...

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Nyquist–Shannon sampling theorem for Quantum Evolution
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3 votes

In order to apply Nyquist-Shannon sampling theory, we need to know the maximum frequency that will be present in the signal we intend to measure. We will do this by rewriting a time-dependent ...

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