Questions tagged [mathematics]

Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.

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Finding a global phase that transform the Hadamard gate to an element of $SU(2)$ and propose an evoultion operator which implents the operation

I was looking back over an old assignment and I came across a question I wasn't quite sure how to do the problem statement is as follows: The Hadamard rotation is an element of the group $U(2)$. (i)...
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given an EPR pair, how do I calculate expectation value?

given A and B share EPR pairs $ (|00⟩+|11⟩)/√2$ both are free to measure their own qubit with the following measurement settings A measures with $[ |0⟩, |1⟩ ]$ B measures with $[ sin(3π/8)|0⟩ + cos(...
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Minimum Multi-Degree Polynomials representing Boolean Functions

In the 10th Anniversary Edition of Nielsen and Chuang Quantum Computation and Quantum Information textbook, Chapter 6.7 talks about Black Box algorithm limits. It is given: $f:\{0,1\}^n \...
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Angular Error associated with Quantum Search Algorithm

Chapter 6.3 of "Quantum Computation and Quantum Information 10th Anniversary Edition" textbook by Nielsen and Chuang talks about using the Quantum Counting Algorithm to find the number of solutions to ...
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What is the HOG test and how would it help proving quantum supremacy?

Proposed experiments in achieving quantum supremacy, such as with BosonSampling or using random circuits, have been described as using a (not necessarily Turing complete) quantum computer to perform ...
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20 views

What is the probability that measurement finds it in the $|0\rangle$ state?

Suppose that there is an ensemble with 60% of the states prepared in $$|a\rangle=\sqrt{\frac{2}{5}}|+\rangle-\sqrt{\frac{3}{5}}|-\rangle$$ and 40% in: $$|b\rangle=\sqrt{\frac{5}{8}}|+\rangle+\sqrt{\...
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Motivation for the definition of k-distillability

Definition of k-distillability For a bipartite state $\rho$, $H=H_A\otimes H_B$ and for an integer $k\geq 1$, $\rho$ is $k$-distillable if there exists a (non-normalized) state $|\psi\rangle\in ...
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Geometric interpretation of 1-distillability

This is a sequel to Motivation for the definition of k-distillability Geometrical interpretation from the definition of 1-distillability The eigenstate $|\psi\rangle$ of the partially ...
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Construction of Phase Shift Operation used in Quantum Search

In Chapter 6 of "Quantum Computation and Quantum Information" Textbook by Nielsen and Chuang, Box 6.1 gives a circuit example of Quantum Search Algorithm done on a two-bit sized search space. The ...
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48 views

Equivalent determinant condition for Peres-Horodecki criteria

The Peres-Horodecki criteria for a 2*2 state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (...
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What is the motivation for Weyl matrices in quantum information theory?

Quantum Entanglement and Geometry — Andreas Gabriel (2010) — Sec: 2.3.4 ~p. 11 Another basis for $d\times d$-dimensional matrices that has proven to be quite useful in quantum information theory is ...
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How are orthogonal sets of pure states arranged in state space?

It is well known that the state of a (pure) qubit can be described as a point on a two-dimensional sphere, the so-called Bloch sphere. The mapping $\lvert\psi\rangle\mapsto \boldsymbol r_\psi$ that ...
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Clarification of a procedure to compute the product of the exponential of two matrices

In trying to understand a method outlined here (page 3, subroutine 1). Consider $$R_3 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} .$$ Let $A$ be a ...
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230 views

Understanding a quantum algorithm to estimate inner products

While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation. Consider ...
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Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

Consider an arbitrary qudit state $\rho$ over $d$ modes. Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation: $$\rho=\frac{1}{d}\left(\mathbb I +\...
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Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere

The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia: Quantum ...
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Purity of mixed states as a function of radial distance from origin of Bloch ball

@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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Method to find $r$ in the case when $r'$ returned by the continued fractions procedure is a factor of $r$

In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the ...
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112 views

What is the spectral decomposition of the Pauli $X$ gate?

The definition of spectral decomposition is as follows: Assume the eigenvectors of $\hat{A}$ define a basis $\beta=\{|\psi_j\rangle\}$. Then $$A_{kj}=\langle\psi_k|\hat{A}|\psi_j\rangle=\alpha_j\...
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56 views

Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem

In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following. $$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$ ...
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Structural Physical Approximation of Partial Transpose

To make the partial transpose a complete positive and therefore physical map, one has to mix it with enough of the maximally mixed state to offset the negative eigenvalues. The most negative ...
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Graphical Calculus for Quantum Circuits

So far I have read a little bit about zx-calculus & y-calculus. From Reversible Computation: The zx-calculus is a graphical language for describing quantum systems. The zx-calculus is an ...
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Rigorous security proof for Wiesner's quantum money

In his celebrated paper "Conjugate Coding" (written around 1970), Stephen Wiesner proposed a scheme for quantum money that is unconditionally impossible to counterfeit, assuming that the issuing bank ...
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Trace distance of two classical-quantum states

I have these two classical-quantum states: $$\rho = \sum_{a} \lvert a\rangle \langle a\lvert \otimes q^a \\ \mu = \sum_{a} \lvert a\rangle \langle a\lvert \otimes r^a $$ Where $a$ are the classical ...
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How does a map being “only” positive reflect on its Choi representation?

We know that a map $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ being completely positive is equivalent to its Choi representation being positive: $J(\Phi)\in\operatorname{Pos}(\mathcal Y\otimes\mathcal ...
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How can I calculate the inner product of two quantum registers of different sizes?

I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand ...
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Direct derivation of the Kraus representation from the natural representation, using SVD

$\newcommand{\Y}{\mathcal{Y}}\newcommand{\X}{\mathcal{X}}\newcommand{\rmL}{\mathrm{L}}$As explained for example in Watrous' book (chapter 2, p. 79), given an arbitrary linear map $\Phi\in\rmL(\rmL( \X)...
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How to measure superposition coefficients to determine state?

There was a problem at the Winter 2019 Q# codeforces contest (that is now over), which I cannot find a mathematical solution for. The problem goes like this: You are given 3 qubits that can be in one ...
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What role do Hecke operators and ideal classes perform in “Quantum Money from Modular Forms?”

Cross-posted on MO The original ideas from the 70's/80's - that begat the [BB84] quantum key distribution - concerned quantum money that is unforgeable by virtue of the no-cloning theorem. A ...
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Can an isometry leave entropy invariant?

Consider two finite dimensional Hilbert spaces $A$ and $B$. If I have an isometry $V:A\rightarrow A\otimes B$, under what condition can I find a unitary $U:A\otimes B\to A\otimes B$ such that $$U\rho_{...
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Writing the transformation matrix for the following in terms of Kronecker products of elementary 2-qubit gates

I have a set of transformations that transforms $|11001\rangle\to |10101\rangle$ which is basically keeping the leftmost qubit as it is and then it is just the CNOT between the successive qubits, I ...
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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 ...
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Square root of CNOT and spectral decomposition of the Hadamard gate

I'm trying to compute the spectral decomposition of the Hadamard gate but I'm making a mistake somewhere. Note: I believe (though I may be wrong so correct me if I am) that spectral decomposition is ...
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Encoding bosonic degrees of freedom

A well-known way of encoding $N$ levels of a harmonic (bosonic) oscillator is as follows: \begin{equation} |n\rangle = |1\rangle^{\otimes n} \otimes |0\rangle^{\otimes N-n+1} \quad,\qquad ...
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How to show a density matrix is in a pure/mixed state?

Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
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Eigenstate of unitary operator used for Order-Finding

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen and Chuang", chapter 5.3.1 introduces the concept of solving the Order-Finding Problem. (Eqn 5.36) states ...
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55 views

Correct way of expressing a measurement in a different computational basis

Sometimes we find that the result we want from a quantum algorithm is expressed in terms of a basis that is different from the usual computational basis, which I will call $$ B_C = \left\{ \lvert 0 \...
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Burnside Decomposition in Kuperberg's Hidden Shift

In "Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem", Kuperberg writes that $\mathbb{C}[G]$ has a "Burnside decomposition" of $$\mathbb{C}[G]\cong \bigoplus_{V}...
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Proof that Grover's operator can be written as $D_N=-H_n R_N H_n$

I am interested in showing the validity of the Grover operator. Now there are several ways to show it. One way is with complete induction. It has to be shown that the following relationship applies: $...
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Are X-state separability and PPT- probabilities the same for the two-qubit, qubit-qutrit, two-qutrit, etc. states?

On p. 3 of "Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures" (https://arxiv.org/abs/1501.02289), it is ...
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Hidden shift problem as a benchmarking function

I encountered the hidden shift problem as a benchmarking function to test the quantum algorithm outlined in this paper (the problem also features here). There are two oracle functions $f$, $f'$ : $...
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How to understand the operators for watermarking schemes?

Note: Cross-posted on Physics SE. I am reading a research article based on quantum image watermarking (PDF here). The authors have defined some unitary transforms for the watermarking schemes, which ...
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Reason for evaluating $a^x \bmod N$ from $x = 0$ to $N^2$

As per the Shor's algorithm, we need to evaluate $a^x \bmod N$ from $x = 0$ to $N^2$. What is the reason for this? Why can't we just evaluate for $N$, $2N$ or something like that?
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Nielsen & Chuang Exercise 2.3 - “Matrix representation for operator products” [closed]

Reproduced from Exercise 2.3 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition): Suppose $A$ is a linear operator from vector space $V$ to vector space ...
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Nielsen & Chuang Exercise 2.5 - Inner products of complex vectors [closed]

Reproduced from Exercise 2.5 of Nielsen & Chuang’s Quantum Computation and Quantum Information (10th Anniversary Edition): A function $(\cdot, \cdot)$ from $V × V$ to $C$ is an inner product if ...
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Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” [closed]

Reproduced from Exercise 2.2 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition): Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\...
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Nielsen & Chuang Exercise 2.4 - “Matrix representation for identity” [closed]

Reproduced from Exercise 2.4 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition): Show that the identity operator on a vector space $V$ has a matrix ...
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379 views

Nielsen & Chuang Exercise 2.1 - “Linear dependence: example” [closed]

Reproduced from Exercise 2.1 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition): Show that $(1, −1)$, $(1, 2)$ and $(2, 1)$ are linearly dependent. ...
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Quantum teleportation of a state, from one of two bases

I'm watching Christian Schaffner's talk on quantum position-based cryptography (link here) and have a question about a particular application of teleportation. At about the 16:40 mark, he seems to ...
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How are multi-qubit gates extended into larger registers? [duplicate]

Implementing a single-qubit gate in a multi-qubit register is relatively easy. For example, this gate: This is equivalent to $I \otimes H \otimes I$. If the $H$ gate was on the first bit, it would be ...