Talking about Quantum Information and Quantum Machine Learning from SWAP Test

How to compare if two quantum states are the same?

Thinking about quantum physics in terms of information reveals many interesting phenomena (even if we leave out the interpretation part for now). First of all, we know that in the world of quantum mechanics, the wave equation requires information to be conserved, which means that all continuous and smooth physical system evolutions are reversible Unitary operations (just speaking of this, there is already a " black hole information paradox " caused by Hawking radiation . , there are many subtle problems), if someone can grasp the entire cosmic wave function from a macroscopic perspective, then the possibility of the future and the past will be in full view. It is a pity that the world is not so simple. The trouble is that we never see the superposition of wave functions in our experiments (for example, when electrons are measured, they are either spin-up or spin-down), whether it is the "collapse" of the wave function interpreted by Copenhagen or more. In decoherence under the world interpretation, there is a process that allows us to read only the eigenstates of the wave function. This means that when we want to extract information from the wave function, we destroy it (at least in a local view), of course through examples like the " quantum erasure experiment " we can re-see the interference fringes (superposition states) by giving up extracting information Characteristics).

Since we will destroy the quantum state in the process of measurement, it is not enough to obtain enough information through repeated measurement (from this point on, assume that we know how to make a specific quantum state, because the " non-replicability theorem " prohibits us from arbitrarily copying unknown quantum states. This is of course possible but extremely uneconomical. After all, it is very difficult to completely reconstruct a wave function ( Quantum State Tomography ) through experiments, because the size of its matrix increases exponentially with the number of qubits (one qubit The element is a 2X2 matrix, and ten qubits are a 1024X1024 matrix). Closer to home, even if it is difficult to completely reconstruct a quantum state, it does not mean that it will be more difficult to compare the degree to which two quantum states are the same. Through the SWAP test , we can simply obtain how similar the two quantum states are without knowing the details of the two quantum states (not It is easier to need information about the entire quantum state).

How to compare whether two quantum states are the same without knowing the quantum state information? This is the ingenious information provided by the entangled state generated by the CWAP gate . We do not need to measure these two unknown quantum states of arbitrary dimensions to obtain their information. We only need to measure an additional ancilla qubit (auxiliary qubit) to obtain the relevant information. information about the overlap of the wave functions (note that we still know nothing about the contents of these two quantum states). Summarize the details with the magic box method: imagine a (SWAP test) black box with an indicator light (ancilla qubit) on it, if we throw two quantum states in exactly the same (complete overlap), the indicator light will be 100% off, If there is no overlap at all (that is to say they are orthogonal vectors), then 50% will light up/50% will not light up, and the state in between is there

The chance indicator will not light up (x is the degree of similarity). So suppose we have two quantum states to compare, then we can know the degree of overlap by measuring the ratio of the light emitted by the indicator light. The complexity will not change as the quantum state increases, and the number of experiments required is constant.

More precisely, the result of the SWAP test will make the input quantum state

, into a symmetric quantum state

(while the ancilla qubit is facing down), and the antisymmetric quantum state

(with the ancilla qubit pointing upwards), the secret of the SWAP test is that by generating symmetric and anti-symmetric entangled quantum states, nature reveals the degree of coincidence of two unknown quantum states. The shock and thinking brought to me by this SWAP test measurement is very similar to quantum teleportation , we do not directly measure the quantum state but can do some interesting things sideways: the former is to obtain the degree of overlap between two quantum states without knowing it, The latter is to transmit its information to a distant place without knowing the quantum state. Are there any uses for the SWAP test measurement other than in fundamental physics? The answer is, of course, yes, in quantum machine learning QML. In quantum machine learning, we will need to compare whether a new vector (data point) is closer to the previous A vector cluster or to the previous B vector cluster, that is, to compare the vector distance and compare the vector distance. It is the core of the QSVM algorithm. SWAP test is an important step for us towards QSVM (that is, classification), allowing us to achieve exponential speedup with significantly low complexity when comparing vector distances! ! !

This article is the theoretical framework behind the experiments that my PhD is doing. This article is written to commemorate the successful upgrade of senior Jaren's doctoral defense to Dr.Jaren in the Dzmitry group. Thanks to the smart boss who came up with such an interesting and non-mainstream research direction, and the senior who has already guided me in the direction :)

The original link is a physics apprentice staring at the singularity