Quantum algorithms running on quantum computers constitute a unique technology. Despite this, they share enough similarities with tensor network algorithms that the two are often treated as rivals. Thought they are partners for 3 reasons: 1. They can deliver value today. 2. They can be used test out and develop quantum algorithms. No supercomputers are required, as they can run on ordinary classical computers.. 3. They will only become more performant when implemented on, or in combination with, quantum computers. This because when the so-called “rank” of a tensor increases, the algorithms get slower on classical computers and quantum computers aren’t affected by this. . ![[Pasted image 20240406154738.png]] #### Analogy to understand Tensor Networks link to Quantum Imagine that people found the answers to their problems by jumping high in the air, plucking a delicate object - the answer - out of the sky before carefully bringing it back down to the ground with them as they land. The answers to the questions that involve more intricate correlations live higher up. These are the extremely valuable answers that everyone wants to find. A tensor network performs every step of this problem-solving procedure very well – much better than alternative classical algorithms. Quantum computers, on the other hand, can jump much higher into the air. Unfortunately, they tend to lose their balance and have a hard time landing. This means they may not find the optimal answer to the problem and when they crash-land, only some fragments of their answer survive intact. To picture it more clearly, imagine that your optimal answer in the sky is a great work of literature but the wreckage of the crash-landing consists of torn pages. The book’s insights were in the connections (or correlations) between the sentences but all you can recover is how frequently various phrases appeared in the text. As classical hardware improves, tensor networks will be able to jump higher but still nowhere near as high as quantum computers. At the same time, the fidelity of quantum computers should improve. This will enable them to jump in a much more controlled and accurate way. Ingenious methods are being developed to prevent quantum computers from losing too much of the information contained in the answer when they land. ### Quantum Tensor Networks Quantum algorithms consist of three main steps:  1. Encoding the data (also known as quantum state preparation),  2. Operating on the data vector (mainly with matrix-vector multiplications)  3. Reading the results (also known as quantum state tomography).  Implementing these steps for arbitrary vectors and matrices requires exponentially large quantum circuits. Without an astute approach, this would make large-scale quantum computing unachievable. Fortunately, tensors networks save the day. They represent data in a very similar way to quantum circuits. When we only consider vectors and matrices that are effectively represented as tensor networks, then all these steps can be performed efficiently: that is, with linear, not exponential, complexity in the number of qubits. ---- Tensor Networks are one of most efficient at simulating quantum circuits. ![[Pasted image 20220118151319.png]] Since it is classical we can save some states, in QC we need to perform all operations in the same ![[Pasted image 20220118151815.png]]