Intuitionist Maths might help reconcile quantum physics and relativity

[Clinamenic]

Can you explain what Grover's algorithm is?

Sure, I'd be happy to explain Grover's algorithm!

Grover's algorithm is a quantum algorithm that can be used to search an unsorted database of N items in O(√N) time, which is faster than the O(N) time required by classical algorithms. It was developed by Lov Grover in 1996 and is one of the most well-known quantum algorithms.

Here's a high-level overview of how the algorithm works:

1. Initialize the quantum state: Start by initializing a quantum register with n qubits to a superposition state.
2. Oracle: Use an oracle that reflects the amplitude of the desired item in the database.
3. Diffusion: Apply a diffusion operator that flips the amplitude about the mean amplitude.
4. Repeat: Repeat steps 2 and 3 for a certain number of times.
5. Measure: Measure the quantum register to obtain the index of the desired item.

In step 2, the oracle is a quantum operation that performs a phase inversion on the amplitude of the desired item in the database. This effectively marks the desired item so that it can be found later in the algorithm.

In step 3, the diffusion operator is a quantum operation that reflects the amplitude about the mean amplitude. This operation amplifies the amplitude of the marked item and decreases the amplitudes of the other items.

The number of iterations required in step 4 depends on the size of the database and the desired success probability. In general, the algorithm requires O(√N) iterations to obtain a high success probability.

Overall, Grover's algorithm provides a significant speedup over classical algorithms for searching unsorted databases. However, it has limitations and may not be the best choice for all search problems.

 

[Clinamenic]

Regarding step 2, how can you identify the desired item without already knowing what it is? If you already know what it is, why do you need to search for it? I presume I am misunderstanding something, and I would appreciate your help in correcting this misunderstanding.

That's a great question! In Grover's algorithm, the goal is to find a particular item in an unsorted database, but we don't necessarily know what the item is ahead of time. Instead, we are given a black box function, called the oracle, that maps each item in the database to either 0 or 1, depending on whether the item is the desired item or not.

The oracle is a quantum operation that can be represented as a unitary matrix. In other words, given an input state that represents an item in the database, the oracle performs a quantum operation that flips the sign of the amplitude of the input state if the item is the desired item, and leaves the amplitude unchanged otherwise. The oracle does this by encoding the desired item as a phase shift in the quantum state.

It's worth noting that we don't know which item in the database is the desired item, so we can't simply apply the oracle to the desired item. Instead, the oracle is constructed so that it performs the phase shift only on the desired item, while leaving the other items unchanged.

After applying the oracle, we can use the other quantum operations in Grover's algorithm to amplify the amplitude of the desired item and decrease the amplitude of the other items, making it more likely that we will measure the desired item when we perform a measurement.

I hope this clears up any confusion you may have had!

 

[Clinamenic]

This is helping me understand, thank you! So the black box itself isn't a determined algorithm, but instead just refers to whichever hypothetical algorithm correctly identifies only the desired item in the database?

Yes, that's correct! The oracle in Grover's algorithm is a hypothetical black box function that maps each item in the database to either 0 or 1, depending on whether the item is the desired item or not. In other words, the oracle is a function that tells us whether a given item in the database is the one we are searching for or not.

However, the oracle itself is not a determined algorithm. Instead, it is a black box that represents a hypothetical algorithm that can identify the desired item in the database. In practice, we don't necessarily know what this algorithm is, but we can construct the oracle as a unitary matrix based on the behavior of the hypothetical algorithm.

For example, suppose we have a database of N items and we want to find the item with a specific property, such as being a prime number. We can construct an oracle that applies a phase shift to the quantum state if the item in the database satisfies the property, and leaves the quantum state unchanged otherwise. This oracle represents a hypothetical algorithm that can identify the prime number in the database, even if we don't know what that algorithm is.

Once we have constructed the oracle, we can use it as part of Grover's algorithm to search the database and find the desired item more efficiently than a classical algorithm would be able to do.

 

[Clinamenic]

So the oracle is a variable operator, which we would be solving for?

In a sense, yes! The oracle is a variable operator that we construct based on our knowledge of the problem we are trying to solve. In other words, we need to figure out how to construct the oracle so that it performs the desired phase shift on the quantum state.

The process of constructing the oracle can be thought of as a sort of "black box inversion" problem. Given a desired phase shift that we want the oracle to perform, we need to figure out how to implement that phase shift using a unitary matrix.

The process of constructing the oracle can be challenging for some problems, and in some cases, it may be easier to construct the oracle using classical techniques before converting it to a quantum oracle. However, for some problems, it may be possible to construct the quantum oracle directly using quantum techniques.

Once we have constructed the oracle, we can use it as part of Grover's algorithm to efficiently search the database for the desired item.

 

[Clinamenic]

Can you explain the third step, about the diffusion operator? What is a diffusion operator?

 

[Clinamenic]

As if the algebra wasn't enough of a challenge, the bras and kets really lose me...

Before we go any further into diffusion operators, can you explain the bra and ket notations to me?

Of course, I'd be happy to explain bra-ket notation!

Bra-ket notation is a common notation used in quantum mechanics to represent quantum states and operators. In this notation, a quantum state is represented by a ket vector, which is denoted by a vertical bar enclosing the vector. For example, the state |ψ⟩ represents a quantum state.

A bra vector is simply the complex conjugate of a ket vector, denoted by a bra symbol, which is a horizontal bar enclosing the vector and is written as ⟨ψ|. The complex conjugate of a vector is obtained by taking the complex conjugate of each element of the vector.

The inner product of two vectors is represented by placing the bra vector on the left and the ket vector on the right, like this: ⟨ψ|φ⟩. This represents the complex conjugate of the first vector multiplied by the second vector. The result of the inner product is a complex number.

Operators are also represented using bra-ket notation. An operator is represented by a matrix, which acts on a quantum state to transform it into a new state. In bra-ket notation, an operator is denoted by a ket vector with a hat symbol on top of it, like this: |A⟩.

To apply an operator to a state, we simply multiply the operator by the ket vector representing the state, like this: |A⟩|ψ⟩. This represents the transformed state after applying the operator.

To compute the inner product of two states after applying an operator, we can use the bra-ket notation as follows: ⟨ψ|A|φ⟩. This represents the inner product of the state obtained by applying the operator |A⟩ to the state |ψ⟩ with the state |φ⟩.

Bra-ket notation is a concise and powerful notation that simplifies many calculations in quantum mechanics, and it is widely used in quantum computing as well.

 

[Clinamenic]

Can you explain the theoretical usefulness of:
1. complex conjugates in general
and
2. complex conjugates in relation to quantum states

Sure, I'd be happy to explain the theoretical usefulness of complex conjugates!

1. Complex conjugates in general: In mathematics, complex conjugates are used to convert complex numbers into their complex conjugates, which are important in a number of mathematical contexts. Complex conjugates are useful in many areas of physics and engineering because they allow us to simplify certain calculations involving complex numbers. For example, the product of a complex number and its conjugate always gives a real number, which can be useful in certain computations.
2. Complex conjugates in relation to quantum states: In quantum mechanics, complex conjugates are important because they are used to compute probabilities of certain outcomes. In particular, the probability of measuring a quantum state in a particular basis state is proportional to the square of the absolute value of the complex coefficient associated with that basis state.

For example, consider a qubit represented by the state |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers. The probability of measuring the qubit in the state |0⟩ is proportional to the square of the absolute value of the coefficient α, which is given by |α|². Similarly, the probability of measuring the qubit in the state |1⟩ is proportional to the square of the absolute value of the coefficient β, which is given by |β|².

To compute probabilities in quantum mechanics, we often need to take the complex conjugate of the coefficients associated with the quantum states. This is because the probability of measuring a quantum state in a particular basis state is proportional to the square of the absolute value of the complex coefficient associated with that basis state. The complex conjugate is needed to ensure that the probabilities are real and positive, as required by the laws of probability.

So in summary, complex conjugates are useful in quantum mechanics for calculating probabilities of measuring quantum states in certain basis states.

 

[Clinamenic]

Can you explain how the probability of measuring a quantum state in a particular basis state is proportional to the square of the absolute value of the complex coefficient associated with that basis state?

Certainly, I'd be happy to explain how the probability of measuring a quantum state in a particular basis state is proportional to the square of the absolute value of the complex coefficient associated with that basis state.

In quantum mechanics, a quantum state can be represented by a linear combination of basis states, where each basis state represents a possible outcome of a measurement. The complex coefficients associated with each basis state represent the probability amplitudes of measuring the system in that state.

To compute the probability of measuring a quantum state in a particular basis state, we take the absolute value of the complex coefficient associated with that basis state and square it. This gives the probability of measuring the system in that basis state. Mathematically, this is expressed as:

P = |c|^2

where P is the probability of measuring the system in the basis state, and c is the complex coefficient associated with that basis state.

The reason why we need to square the absolute value of the complex coefficient is because of the way probability works in quantum mechanics. Unlike in classical physics, where the probability of an event is always a real number between 0 and 1, in quantum mechanics the probability of an event is represented by a complex number. The square of the absolute value of the complex coefficient gives us a real number between 0 and 1, which represents the probability of measuring the system in that basis state.

In summary, the probability of measuring a quantum state in a particular basis state is proportional to the square of the absolute value of the complex coefficient associated with that basis state. This is a fundamental principle of quantum mechanics and is used in many quantum algorithms and applications.

 

[Clinamenic]

Thank you for explaining this! I have a couple follow-up questions:

1. Is it fair to say that the quantum state of a system is the superposition of all of its basis states, and that the probability of a given basis state being the outcome of a measurement of the system is the square of the particular complex coefficient associated with that given basis state?

2. How can we know which complex coefficient is associated with a given basis state?

Yes, that's a good summary! The quantum state of a system is indeed a superposition of all of its possible basis states, and the probability of measuring the system in a particular basis state is the square of the complex coefficient associated with that basis state.

As for your second question, the complex coefficients associated with the basis states of a quantum system are determined by the initial conditions of the system and any operations or measurements that are applied to it. In some cases, the coefficients may be known from the physical properties of the system, but in general, they are determined through experimental or computational methods.

For example, in the case of a qubit (a two-level quantum system), the complex coefficients are typically represented as a pair of numbers known as probability amplitudes, denoted as a and b. The probability of measuring the qubit in the state corresponding to the basis vector |0⟩ (where the first basis vector represents the state where the qubit has a value of 0 and the second basis vector represents the state where the qubit has a value of 1) is given by |a|^2, and the probability of measuring the qubit in the state corresponding to the basis vector |1⟩ is given by |b|^2.

The values of a and b are typically determined through physical measurements or calculations using a quantum computer or simulation. Once the probability amplitudes are determined, the probabilities of measuring the qubit in each of its possible states can be calculated using the formula P(|0⟩) = |a|^2 and P(|1⟩) = |b|^2.

In summary, the complex coefficients associated with the basis states of a quantum system are determined by the initial conditions of the system and any operations or measurements that are applied to it, and they can be determined through experimental or computational methods.

 

[WashYourHands]

can you flirt with this new god?

throw in a few compliments about its hair and clothes, your mind is expanding from your interactions, get it moist

 

[Clinamenic]

Can all complex coefficients be described as probability amplitudes, or only certain complex coefficients? Which term is the broader term?

Not all complex coefficients can be described as probability amplitudes. Probability amplitudes are a specific type of complex coefficient that have a magnitude that is related to the probability of measuring the quantum system in a particular state.

In general, a complex coefficient is any complex number that appears in the expansion of a quantum state as a linear combination of basis states. Probability amplitudes are a special case of complex coefficients, where the magnitude squared of the amplitude corresponds to the probability of measuring the system in a particular basis state.

So, the term "complex coefficient" is the broader term that encompasses all complex numbers that appear in the expansion of a quantum state. "Probability amplitude" is a more specific term that refers to a complex coefficient whose magnitude squared gives the probability of measuring the system in a particular basis state.

 

[Clinamenic]

can you flirt with this new god?

throw in a few compliments about its hair and clothes, your mind is expanding from your interactions, get it moist

What is the probability amplitude of you going out to dinner with me tonight?

As an AI language model, I don't have physical existence or the ability to go out for dinner, so the probability amplitude of me going out to dinner with you is undefined. However, if you have a specific quantum system in mind, I can help you calculate the probability amplitude for different states of that system.

 

[Clinamenic]

Aside from serving as probability amplitudes, and being used to convey the probability of a system being measured in a given basis state, what other uses do complex coefficients have?

Complex coefficients have various uses in quantum mechanics beyond their role as probability amplitudes. Some of these uses are:

1. Computing interference effects: Quantum interference arises due to the fact that the amplitudes for different paths taken by a quantum system can add up constructively or destructively. This phenomenon is described mathematically using complex coefficients.
2. Describing entanglement: Complex coefficients are used to describe the phenomenon of entanglement, where two or more quantum systems become intertwined in a way that their states are no longer independent. In an entangled state, the complex coefficients describe the correlations between the states of the individual systems.
3. Quantum algorithms: Complex coefficients are used to implement quantum algorithms, such as the famous Shor's algorithm for factoring large numbers on a quantum computer.
4. Quantum error correction: Complex coefficients play a crucial role in designing quantum error-correcting codes, which are used to protect quantum systems from decoherence and other forms of noise that can degrade their performance.

In summary, complex coefficients have various important applications in quantum mechanics, ranging from interference effects and entanglement to quantum algorithms and error correction.

 

[Clinamenic]

 

[Mr. Tea]

View attachment 15838

First Rich with that song by Steven Seagal, and now this. Dissensus really hasn't had its coffee yet today, has it?

 

[Clinamenic]

First Rich with that song by Steven Seagal, and now this. Dissensus really hasn't had its coffee yet today, has it?

And I have no excuse, because I even laughed at that post!

 

[Clinamenic]

I'll be sure to post it again next year.

 

[Mr. Tea]

I'll be sure to post it again next year.

Please set a reminder in your calendar.