Questions tagged [fidelity]

For questions about the fidelity between quantum states.

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Fidelity and Uhlmann's theorem in the context of source coding

In quantum source coding, we have an encoder $\mathcal{E}$ and a decoder $\mathcal{D}$ which are some quantum channels. Given a state $\rho_A$ on Hilbert space $\mathcal{H}_A$, we wish to encode and ...
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Calculating state fidelity and space complexity of Minimum Eigen Optimizers (VQE, QAOA and Grover Optimizer) in qiskit

I'm a beginner in using Qiskit and my Computer Science background is not extensive. But I understand the Quantum Physics aspects of it relatively well. I solved a QUBO problem in Qiskit using VQE (...
3 votes
1 answer
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Does a fidelity of $\mathcal{F}(U_1|0\rangle, U_2|0\rangle)=1$ imply that $U_1=U_2$?

I'm now studying quantum ML and now studying about fidelity ($\mathcal{F}$). To my knowledge, fidelity means the distance between two quantum states, $\textit{i.e.,}$ if $\mathcal{F} ==1$, then the ...
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8 votes
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Prove that a channel is close to acting on only one system

Background Suppose I have a quantum channel $\Phi:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_1)\otimes B(\mathcal{H}_2)$, such that there is some small $\epsilon$ such that for any two input states $\...
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Are there any systematic tools for estimating expected error rate?

Like we all probably know, today’s NISQ computers are, as their name implies - noisy, i.e high error rate is involved when using them. Hence, if we desire valuable results then we should come up with ...
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What is success rate and how to calculate it?

This paper describes noise adaptive qubit routing that aims to increase program success rates. This method also implemented in Qiskit routing method. However, I am confused about the term success rate ...
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What is fidelity in quantum computing?

As I studied quantum computing, I saw term 'Fidelity' in many papers that related to quantum algorithm. So, I really wonder about following two things. What is the real meaning of 'Fidelity' (As I ...
3 votes
1 answer
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How to calculate the average fidelity of Pauli error channel

This question is related to this and this. I am working on Qiskit to design QEC schemes. My model works with Pauli errors. I would like to give to my Pauli error channel probabilities $p_x,p_y,p_z$ ...
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Commutative operators

I have got a 2-qubit circuit with the following instructions: ...
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which hardware platform is best for single qubits?

Please share a paper most recent data on single qubit hardware comparisons. Mainly gate single qubit gate fidelities and coherence times.
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Why is the fidelity, rather than the trace distance, the standard choice to compare quantum states?

I don't think it's particularly controversial to say that the "standard" way people use to compare quantum states is via the fidelity. Yes, sometimes the trace distance is used as well, but ...
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What makes dynamical decoupling a good method since the fidelity after using it can only reach 0.85 or so?

From this paper, I see that the fidelity of single qubit gate after using dynamical decoupling only reach around 0.85 while I normally saw experiment papers state their fidelity can reach around 0.99 ...
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How to interpret the measurement $\mu$ giving a fidelity equal to the Bhattacharyya coefficient: ${\rm F}(P,Q)={\rm B}(P,Q|\mu)$?

(Bhattacharyya coefficient vs state fidelity) Given two vectors $u,v$ with nonnegative entries, their Bhattacharyya coefficient is $$\mathrm B(u,v)\equiv \sum_a \sqrt{u_a v_a}.$$ Given two positive ...
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Is (square root) fidelity strictly concave?

We know that (square root) Fidelity which is defined as $\text{F}(\rho,\sigma) = \| \sqrt{\rho} \sqrt{\sigma} \|_1 = \text{Tr}(\sqrt{\sqrt{\sigma} \rho \sqrt{\sigma}})$ is satisfies the property of ...
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Scaling of magic state distillation with single & two-qubit gate error

How does the resulting error in magic state distillation scale with the fidelity of single qubit gates and two qubit gates? In most experimental systems, two-qubit gate errors are much larger than ...
3 votes
1 answer
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Game formulation of Quantum GAN

Quantum Generative Adversarial Network (QuGAN) generates a desired quantum state via a minimax game between generator and discriminator (equivalently, it's optimizing a trace distance between ...
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Problem with fidelity value range obtained from this python code

I'm calculating for instance the fidelity between two pure two qubit state, using the following python code: ...
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What use cases are there for 127 qubit QPUs?

IBM have recently announced their 127 qubit Eagle processor. Other approaches, such as Rydberg arrays, have now 256 qubits, as for example in QuEra's QPU QPU. While these are without a doubt ...
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Is it overestimating noise in a QPU if we use infidelities and also quantum channel such as depolarising or amplitude damping?

I have mainly seen two ways of studying noise in quantum algorithm simulation. The first one is to suppose that your quantum gate can be implemented with a probability of success of $\mathcal{F}$ (...
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Qubit fidelity of DWAVE device

Since DWAVE quantum device is constructed using superconducting flux qubits, each qubit cannot be produced identically so that the fidelity of the qubit must be different. DWAVE only provides the ...
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1 answer
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Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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Properties of the generalized fidelity for subnormalized states

The generalized fidelity for quantum states that may be sub-normalized is given by (Defn 3.12) $$F_{*}(\rho, \tau):=\left(\operatorname{Tr}|\sqrt{\rho} \sqrt{\tau}|+\sqrt{(1-\operatorname{Tr} \rho)(1-\...
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What scheme should be used in case of applying non-Cliffords to estimate probability of success?

For Clifford gates (when performing randomized benchmarking and starting from ground state) the final state is always ground. It is acquired by applying at the end recovery gate, which transfers the ...
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How to calculate the threshold for gate fidelity?

I've been interested in gate fidelity lately. In the meantime, I came up with a specific example. Let's think about the encoding of situation $\alpha_0 |0\rangle + \alpha_1 |1\rangle \rightarrow \...
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How can I get fidelity of a gate from randomized benchmarking?

I found (page 33) the method of finding fidelity from fit by "interleaved and reference decay" according to the sequence fidelity formula: $$F_{ref}=Ap_{ref}^{m}+B,$$ where $p_{ref}^{m}$ is ...
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Amplitude encoding: distinction between negative and positive real values

I want to encode two vectors into qubits and compute a distance between them that corresponds/is proportional to the euclidian distance of the real vectors. The encoding I use is A) qiskit ...
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How large can we make the fidelity between mixed states by allowing unitaries?

For pure states, it is known that one can always find a unitary that relates the two i.e. for any choice of states $\vert\psi\rangle$ and $\vert\phi\rangle$, there exists a unitary $U$ such that $U\...
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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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1 answer
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What are min and max overlaps of a maximally entangled state with a separable state?

Let $A,B$ be Hilbert spaces of dimension $d$. Let $\rho$ be some separable quantum state of the composite system $AB$. Given a maximally entangled state: $$\vert\phi\rangle = \frac{1}{\sqrt{d}}\sum_{i=...
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Approximate Cloning

Question Consider two single qubit states $\left\{|\alpha_0\rangle,|\alpha_1\rangle\right\}$ which are not orthogonal or parallel, i.e. $\left|\langle\alpha_0|\alpha_1\rangle\right|\ne0,1$. ...
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3 answers
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How are eavesdroppers detected when using BB84 in the presence of noise?

I would like to expand upon this question: What is the probability of detecting Eve's tampering, in BB84? Let's say that when the receiver (colloquially referred to as Bob) receives a qubit and ...
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Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
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What is the connection between Bures metric and (finite) Bures distance?

The Wikipedia page discussing the Bures metric introduces it as the Hermitian 1-form operator $G$ defined implicitly by $\rho G+G\rho = \mathrm d\rho$, and which induces the corresponding Bures ...
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What is a stabilizer state?

I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'. "The number of repetitions depends on the ...
5 votes
2 answers
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Clarification on Watrous' proof of Alberti's theorem on the fidelity function

I am reading John Watrous' quantum information theory book. In the proof of Theorem 3.19 (practically the Alberti's theorem on the characterization of the fidelity function) he claims the following ...
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If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?

Let $\rho = \sum_i \vert i\rangle\langle i\vert \otimes \rho_i$ and $\sigma = \sum_i\vert i\rangle\langle i\vert\otimes\sigma_i$ where we are using the same orthonormal basis indexed by $\vert i\...
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How to prove generalized Uhlmann's theorem?

I think the Uhlmann theorem should be in general of this form: Let $\rho$ and $\sigma$ be density operators acting on $A$, with Schmidt degrees at most $r$, and let $B$ be another Hilbert space with ...
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Trace distance between mixed state and pure state vs trace distance between their purifications

Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that $$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$ where $\|A\...
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How do the extra energy levels of a transmon qubit affect computation/fidelity?

I was reading about transmon qubits, and I know that they are not true two-level systems. Are there any math/papers which talk about how those extra energy levels affect the computation? I'm assuming ...
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How does $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$ [duplicate]

From Quantum Information Theory by Mark Wilde, pg 243 asks to show that $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$, which is described as the Bhattacharyya overlap, or classical fidelity, between ...
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Given $\rho,\sigma$, bound $\min F(\bar\rho,\sigma)$ over $\bar\rho$ such that $F(\bar\rho,\rho)\ge1-\epsilon$

Let $\rho, \sigma$ be states such that $$F(\rho,\sigma) = \delta >0,$$ where $F(\rho,\sigma) = \|\sqrt{\rho}\sqrt{\sigma}\|_1$. Now consider all possible states $\bar{\rho}$ such that $F(\bar{\rho},...
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How to find the distance between a given $\rho$ and the nearest pure state(s)?

I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state: $$ \min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
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What is the best quantum process tomography method?

This question is somewhat related to this question. What is currently the best method for quantum process tomography? By best I mean, the one that can achieve the best accuracy of estimation per qubit ...
2 votes
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What is the best method for estimating average channel fidelity?

This thesis shows an efficient way to estimate average channel fidelity (in chapter 4). However, it is somewhat old (from 2005). Are there any better methods out there? By better I mean: are there ...
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Can Gate Set Tomography work on Quantum Channels?

I stumbled across a new paper on gate set tomography. Can gate set tomography be applied to a quantum channel or multiple quantum channels? Will the same advantages still apply of not having to 'rely ...
3 votes
1 answer
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Can quantum error correction work on any type of channel?

It says on wikipedia that quantum error correction can (at best) correct phase flips and bit flips. A popular form of representing a quantum channel is in its Kraus representation (scroll down to ...
5 votes
1 answer
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How to calculate the average fidelity of an amplitude damping channel

An answer to this question shows how to calculate the average fidelity of a depolarizing channel. How would one go about calculating this for an amplitude dampening channel? I tried working out the ...
4 votes
1 answer
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How to limit the error probability in large scale quantum computers

I am quite stumped by the fact that the roadmaps for quantum computers as given by IBM, Google, IonQ, etc. seem to imply a linear/exponential growth in the size of their quantum computers. Naively, I ...
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How is the connection between Bures fidelity and quantum Fisher information derived?

I recently came to know that there is a connection between Bures Fidelity $(F_B)$ and Quantum Fisher Information $(F_Q)$ given by $$[F_{B}(\rho, \rho_\theta)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + ...
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What is the relation between fidelity and concurrence for a two qubit maximally mixed state?

I am trying to understand the relation between Fidelity and Concurrence for a two qubit maximally mixed state. When I calculate the Fidelity and Concurrence, I observe that Concurrence is zero whereas ...
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