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I read in a book that (https://hub.packtpub.com/quantum-expert-robert-sutor-explains-the-basics-of-quantum-computing/)

160 qubits (quantum bits) could hold $2^{160} \approx1.46\times 10^{48}$ bits while the qubits were involved in computation."

How does this calculation come about?

The context of the statement is that a caffine molecule would require $10^{48}$ bits to be represented by a classical computer. However a quantum computer would require 160 qubits and is thus well suited for such representation.

If I look at this question on Quora, a 512 bit computer (which I suppose are real) would give a largest 155 digit number (https://www.quora.com/How-many-digits-are-in-a-512-bit-number). Isn't that big enough to represent atoms, molecules etc.?

Manu Chadha
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If you have $n$ bits you can combine them in $2^n$ different bit string (this come from combinatorics). Now take $n$ qubits. As any qubit can in superposition of two state, i.e. 0 and 1, $n$ qubits can be in superposition representing all $2^n$ possible bit strings.

The notion that $n$ qubits can hold $2^n$ classical bits is unfortunately misleading because when you measure the qubits, they will collapse to one particular state. This means that information content of $n$ qubits is $n$ classical bits.

Adam Zalcman
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Martin Vesely
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  • Thanks. I am unfortunately still unable to comprehend why a classical computer needs 10^48 bits while a quantum computer will need only 160 bits. Would you know something about that? – Manu Chadha Jan 25 '21 at 11:11
  • This has to do with molecular simulation algorithms, I am no expert in this area, but Quantum Computers allow for easy molecular simulation, like they do Hamiltonian Simulation (it is the same thing), the Quantum computer is not 'magically more powerful' than a classical one, but because of the computational constructs available in Quantum Computing, the algorithms for molecular simulation scale linearly, and for classical computers they scale exponentially. – Bertrand Einstein IV Jan 25 '21 at 19:19
  • This is a subtle concept because Quantum Computers are not just 'exponentially more powerful', but certain problems have Quantum Solutions that are exponentially faster than their Classical Solutions (there are many more nuances, and Quantum Complexity is a field in its own right, but this is the basic idea) – Bertrand Einstein IV Jan 25 '21 at 19:24
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    @ManuChadha: Number of all bits combinations needed for the simulation is $2^{160} \approx 10^{48}$, so you need as many memory places. However, on quantum computer you can save all these combination into superposition state of 160 qubits. – Martin Vesely Jan 26 '21 at 09:35
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    A bit represents a state. On its own, 1 or 0. 2 possible states. 2 bits, 00, 01, 10, 11. 4 states. 3 bits, 000, 001, ..., 111. 8 states. The number of possible states doubles every time you add a bit. Classically, each single state needs to be modeled explicitly by its own bit string (e.g. 1010110). In a quantum computer we can have a superposition of qubits, say, 3 qubits that models all of the (8) possible bit strings. This is done without having to explicitly assign 8 bits to each bit string. Instead, we have a probability (amplitude) associated with each bitstring modeled by just 3 qubits. – Greenstick Jan 26 '21 at 19:00
  • Revising my primitive computer architecture concepts - 2^160 is approx. same as 10^48 (martin), so I need a 160 bit address bus and a RAM of size 2^160 bits? But on Qubit, I just need 160 bit RAM (some special Qubit RAM I suppose) because each bit can represent 2 states simultaneously? Am I correct that current PCs are 64 (2^64 RAM max) and scaling this to 2^160 is the challenge in current PC architecture? With Qubits, I just need a 8 bit address bus to address (2^8 > 160) locations which makes quantum computers better – Manu Chadha Jan 27 '21 at 12:22
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    @ManuChadha: The problem is not in the bus width but memory size. $10^{48}$ bits is $10^{36}$ Tb (terra bits) and this is impossible to achieve. However, with quantum computer you need only 160 qubits in superposition to represent the simulation input. Of course, in the end you are left with only 160 classical bits representing the result (i.e. the optimum) but this is the case for classical computer too. – Martin Vesely Jan 27 '21 at 13:54
  • Got it (I think :)). Not sure if this is a valid question w.r.t. quantum computers, how much memory does 160 qubit computer have? – Manu Chadha Jan 27 '21 at 15:28
  • @ManuChadha: Currently there is no practical implementation of quantum memory (qRAM). There is simply 160 qubits in the processor itself. qRAM is something scientists and engineers are working on very hard. Just note that this conversation is going to be long. I would recommend to post a next question as new one. – Martin Vesely Jan 28 '21 at 08:01