In this paper, there is a statement in the abstract:

Our construction uses $3n + 0.002n \log(n)$ logical qubits, $0.3n^3 + 0.0005n ^3\log(n)$ Toffolis, and $500n^2 +n^2 \log(n)$ measurement depth to factor n-bit RSA integers.

The title of the paper states that 20.000.000 qubits are used to crack RSA-2048 where this presentation -also refers that paper- includes table in pg.22 that maps RSA-2048 to 6189 qubits.

My question is: Which one is the quantity that should be considered for following development in quantum computers? In other words, what is the necessary number of qubits for quantum computers to crack RSA-2048 according to this paper? 6189 or 20.000.000?

Also, definitions of logical qubit, noisy qubit, measurement depth and Toffoli can be very useful for understanding this concept.

  • $\begingroup$ Main question is: which one will factor 2048-bit integer according to this work? 6189-qubit or 20.000.000-qubit quantum computer? $\endgroup$
    – NB_1907
    Commented Jan 19, 2022 at 10:39
  • $\begingroup$ Section 2.4 of the paper talks about `switch from the usual representation of integers to the coset representation of modular integers. and see en.wikipedia.org/wiki/… $\endgroup$
    – kelalaka
    Commented Jan 19, 2022 at 11:24
  • $\begingroup$ I am trying to understand general concept by just focusing on number of qubits. What represents 6189 and 20.000.000? Can we say 20.000.000 noisy qubit corresponds 6189 abstract qubits? And which one corresponds the number that declared by e.g. IBM as number of qubits of quantum computer. $\endgroup$
    – NB_1907
    Commented Jan 19, 2022 at 12:03
  • 2
    $\begingroup$ What is the difference between a physical and a logical qubit? $\endgroup$
    – kelalaka
    Commented Jan 19, 2022 at 14:12
  • 1
    $\begingroup$ Understanding (theoretical) computing power of quantum computers $\endgroup$
    – kelalaka
    Commented Jan 19, 2022 at 14:58

1 Answer 1


The 20,000,000 is the number of physical qubits of a certain quality required and corresponds most closely to the number of qubits quoted by those engineering teams current developing quantum devices. However, quantum computational capability is not just dependent on the raw number of qubits available. The 20,000,000 qubits quoted need to be able to execute a quantum computational gate in 1 microsecond with 99.9% accuracy, interact with a large number of neighbouring qubits and maintain a quantum state for several hours. How close the different devices are to achieving this specification will vary and one has to delve into the details. It is possible that engineers will be able to produce qubits that perform better than this specification in which case, fewer will be required.

A logical qubit should be thought of as an idealised computational resource that executes gates with perfect fidelity, can communicate freely with other logical qubits and can maintain its quantum state indefinitely. The 6189 logical qubits required are not really possible to reduce with improved engineering, but may be reducible via improved algorithmics.

Certain capabilities of logical qubits can be emulated by collections of physical qubits by using quantum error correction codes to correct errors in gate execution and loss of information over time. These noisy/physical qubits can be realised in different ways (most of the major engineering projects use super-conducting qubits), all of which have limitations that may be improved by engineering. The number of physical qubits required to emulate a logical qubit for the duration of the algorithm depends on the quality of the physical qubits. The emulation itself contributes to the computational burden.

Measurement depth is the longest path of gates that quantum information has to flow through to execute the algorithm. The algorithmic complexity will depend both on the number of qubits and the measurement depth. The product of the two is a rough overall measure of this complexity.

A Toffoli gate is a basic sort of gate that enables very general quantum circuits to be built (analogous to how Shannon's theorem allows us to build general computational circuits from NAND gates). From an engineering perspective it is usually the most difficult basic gate to implement and so the number of Toffoli gates is another measure of the engineering challenge. On classical data a Toffoli gate sends three input bits $(a,b,c)$ to three output bits $(a,b,c\oplus a\cdot b)$.


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