Depends!!

I got taken up with the accuracy question on Phase estimation. Below, text-book code

for one qubit with a phase and three h=ancillary cubits. Qubit 3 is the target. Even with

trials up to 8192, still stuck with an estimate of .375 in a situation where .333 is the actual

phase.

Gemini:

In the context of quantum phase estimation, the histogram you've provided represents the probability distribution of the estimated phase angles. Each bar in the histogram corresponds to a particular bitstring, which, when interpreted as a binary number, gives us an estimate of the phase.

Calculating the Estimated Phase:

1. Binary Representation:

   • Each bitstring on the x-axis represents a binary number. For example, "000" corresponds to 0, "001" corresponds to 1, and so on.

2. Phase Angle Calculation:

   • The phase angle is calculated by dividing the binary number by the total number of bits used in the estimation. In this case, let's assume there were 3 qubits used for estimation.

   • For instance, if we consider the bitstring "101", which corresponds to the decimal number 5, the estimated phase angle would be:

     Phase Angle = 5 / (2^3) = 5/

...

The actual phase angle might not be exactly equal to the estimated value due to the inherent randomness of quantum measurements and potential errors in the estimation process.

The accuracy of the estimation depends on factors like the number of qubits used, the quality of the quantum hardware, and the specific quantum algorithm employed.

Qiskit quantum phase estimation; help me understand why the target qubit gets...

                                                         *     *     *

COPILOT

Quantum Computing: could I have an oveview of this paper: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.109.042412

Sure! Here is a brief overview of the paper:

Quantum Phase Estimation: The paper discusses algorithms for estimating the unknown strength of a unitary evolution, which is crucial for quantum computing and quantum metrology.

Coherence-Based Algorithm: It introduces a new coherence-based phase-estimation algorithm that achieves optimal quadratic scaling in mean absolute error and mean squared error.

Adaptive Bayesian Protocol: The algorithm’s optimality is due to its adaptive nature, using a Bayesian protocol to iteratively determine each step based on previous results.

Noise Resilience: The algorithm performs well even in the presence of noise, approaching the theoretical lower bound of errors.

If you have any specific questions about the paper, feel free to ask!

Reference: perplexity.ai

 

A Bayesian protocol is a method used in various fields, including quantum computing, to update the probability estimate for a hypothesis as more evidence or information becomes available. It is based on Bayes’ theorem, which describes how to update the probabilities of hypotheses when given new data.

Here’s a simplified breakdown:

Initial Hypothesis: Start with an initial hypothesis or prior belief about the system or parameter you are studying.

Gather Data: Collect new data or evidence related to the hypothesis.

Update Belief: Use Bayes’ theorem to update the prior belief with the new data, resulting in a posterior belief. This process involves calculating the likelihood of the new data given the hypothesis and adjusting the prior belief accordingly.

Iterate: Repeat the process as more data becomes available, continually refining the hypothesis.

In the context of quantum computing, a Bayesian protocol can be used to iteratively refine estimates of quantum states or parameters, making it a powerful tool for tasks like quantum phase estimation. The adaptive nature of the protocol allows it to efficiently converge to accurate estimates even in the presence of noise.

What does this protocol rely on to change course from one observation to the next?

Key Elements of a Bayesian Protocol

A Bayesian protocol is a systematic approach that allows for the adaptation and refinement of beliefs based on new observations. It consists of several key components:

1. Prior Distribution
   The prior distribution represents our initial beliefs about a parameter or state before any new data is observed. It encapsulates what we know or assume about the parameter based on previous knowledge or assumptions.
2. Likelihood Function
   The likelihood function quantifies the probability of observing the new data given a specific hypothesis or parameter estimate. It measures how likely the observed data is under different possible values of the parameter, providing a way to assess the fit of our model to the data.
3. Posterior Distribution
   Once new data is observed, we update our prior distribution using the likelihood function to obtain the posterior distribution. This posterior distribution reflects our updated beliefs about the parameter after considering the new evidence and serves as the new prior for subsequent iterations.
4. Bayes' Theorem
   Bayes' Theorem provides the mathematical foundation for updating our beliefs. The theorem can be expressed as follows:
   $$P(\theta \mathrm{\mid }D)=\frac{P(D\mathrm{\mid }\theta )P(\theta )}{P(D)}$$
   In this equation:
   • $P(\theta \mathrm{\mid }D)$ represents the posterior probability of the parameter $\theta$ given the observed data $D$.
   • $P(D\mathrm{\mid }\theta )$ is the likelihood of observing the data $D$ given $\theta$.
   • $P(\theta )$ denotes the prior probability of $\theta$.
   • $P(D)$ is the marginal likelihood of the data, which acts as a normalizing constant.
5. Iterative Process
   The Bayesian protocol operates as an iterative process. With each new observation, we refine our estimate of the parameter. The posterior distribution obtained from one iteration becomes the prior distribution for the next iteration, allowing us to continuously adapt our beliefs based on accumulating evidence.

By consistently updating our understanding of a parameter with each new piece of data, the Bayesian protocol enhances the accuracy and reliability of estimates over time. This adaptability makes it particularly valuable in dynamic and uncertain environments, such as those encountered in quantum computing.