To address whether a quantum computer with one fewer logical qubit than required can make progress in factoring an integer using Shor's algorithm, we analyze the qubit requirements and the algorithm's structure:

Qubit Requirements in Shor's Algorithm

Shor's algorithm requires logical qubits for three main components:

1. Control Register (A): Stores the exponent ( a ), typically chos...

Short answer Yes. Shor’s algorithm admits substantial time–memory (depth–qubit) tradeoffs. There isn’t a single hard qubit threshold beyond which you suddenly can factor and below which you can’t. If you are “one qubit short” of a particular circuit layout, you can often recompile to save that qubit at the cost of longer runtime. However, there is a fundamental floor: you need on the order of n lo...

Short answer
• Below about n logical qubits, where n = ⌈log₂N⌉ is the bit-length of the integer N you want to factor, Shor’s algorithm cannot even be started.
• Once you have those n qubits (plus 1 or 2 extra for control), you can reduce the additional work space arbitrarily far by paying with more and more run-time; this gives a family of time–memory trade-offs above the n-qubit threshold...

In classical cryptanalysis one is so used to “I’m 1 GB short so I spill half of my state to disk and pay a factor‐of‐2 slowdown” that it is tempting to ask the same about a quantum computer: “If I’m just one logical qubit shy of the textbook Shor–machine for an n‐bit modulus, am I dead in the water, or can I somehow trade time for that missing piece of memory?”

The short answers are:

1. You ca...

Short answer
Yes, you can trade time for memory in Shor’s algorithm, but only down to a very sharp limit.
• If you still have at least n + O(1) logical qubits (where n ≈ log2 N is the bit-length of the integer you want to factor) you can always re-write the circuit so that it works; it just runs slower.
• If you have even one qubit less than that lower bound you cannot make any further progr...

Excellent question. You've hit on a crucial point in the practical application of quantum algorithms. The answer is a resounding yes, a quantum computer with insufficient qubits can still make progress, and there are direct time-memory tradeoffs available for Shor's algorithm.

A machine that is one qubit short is far from useless; it can still factor the integer, but it will take longer. Let'...

This is an excellent question about the fundamental nature of quantum algorithms and their resource requirements. The short answer is that Shor's algorithm is quite rigid in its qubit requirements, and having "almost enough" qubits doesn't help much - but there are some nuances worth exploring.

Why Shor's Algorithm is Inflexible

Shor's algorithm requires approximately 2n qubits to factor an n...

Background on Shor's Algorithm

Shor's algorithm factors ...

Time-Memory Tradeoffs in Shor's Algorithm

Yes, time-memory tradeoffs are possible in Shor's algorithm, allowing factorization with fewer qubits than the standard implementation requires, though at the cost of increased runtime or classical processing.

Standard Qubit Requirements

For factoring an n-bit integer N, the standard implementation of Shor's algorithm requires:

• 2n qubits for the ...