Sim memory
16 PB
Math & numbers · Frontier
Quantum
Quantum
calculator
Four numbers that frame every quantum conversation: how big 2ⁿ really is, what simulating it classically costs, how many physical qubits error correction demands, and how far a circuit gets before noise wins.
2ⁿstate space · error correction · noise
2⁵⁰ amplitudes
1,125,899,906,842,624
Physical qubits
62,500
Circuit success
36.8%
dim = 2ⁿ · · · memory = 2ⁿ × 16 B · · · physical ≈ n · 2d² · · · success = Fᴳ
Why fifty qubits is a supercomputer problem
Every extra qubit doubles the state space. A register of n qubits lives in a space of 2ⁿ complex amplitudes — at n = 50 that is about 10¹⁵ numbers, and by n = 300 it exceeds the count of atoms in the observable universe. This tool makes that growth concrete, exactly where it can: powers of two up to 2⁶⁴ are shown as exact integers, and beyond that in scientific notation.
Simulating it classically
memory = 2ⁿ × 16 bytes (two float64 per amplitude)
n = 30 → 17 GB · n = 40 → 17.6 TB · n = 50 → 18 PB
n = 30 → 17 GB · n = 40 → 17.6 TB · n = 50 → 18 PB
The memory line is why quantum advantage claims centre on the mid-40s to 50s of qubits: a state-vector simulation must hold every amplitude, and each one costs 16 bytes. A laptop taps out around n ≈ 30; the largest classical machines reach the mid-40s. Above that, exact simulation stops being an engineering problem and becomes an impossibility of arithmetic.
The error-correction bill
surface code: physical per logical ≈ 2d²
d = 25 → 1,250 physical per logical qubit
d = 25 → 1,250 physical per logical qubit
Today's hardware is noisy, so useful machines spend qubits on error correction. In the surface code — the leading approach — one logical qubit costs roughly 2d² physical qubits at code distance d, and distances around 25 are the scale usually quoted for meaningful algorithms: fifty logical qubits then demand tens of thousands of physical ones. The success line completes the picture: with per-gate fidelity F, a G-gate circuit finishes cleanly with probability about Fᴳ, which is why 99.9% fidelity still buys only a few thousand gates — and why error correction exists at all. Related tools: the number-base converter for the binary underneath, and the AI compute calculator for the other frontier's arithmetic.
Common questions
Quantum FAQ
Because the classical description must track an amplitude for every basis state, and n qubits have 2ⁿ of them. Each amplitude is a complex number — two 64-bit floats, 16 bytes — so memory is 2ⁿ × 16 bytes. One more qubit doubles the bill; ten more multiply it by 1,024.
State-vector simulation of about 30 qubits needs ≈17 GB, so a well-equipped laptop manages the high twenties. Large clusters have pushed exact simulation into the mid-40s, where memory reaches tens of petabytes. Beyond that, only approximate or specially structured circuits can be simulated at all.
Distance d measures how many physical errors a code can absorb before a logical error slips through — bigger d, better protection. A distance-d surface-code patch uses roughly 2d² physical qubits (data plus measurement qubits), so protection grows quadratically in cost: d = 25 means about 1,250 physical qubits per logical one.
As a first-order model, yes: if each gate succeeds with probability F independently, G gates all succeed with probability Fᴳ. Real devices add correlated noise, idle errors and measurement error, so the true number is somewhat worse — which makes Fᴳ a fair optimistic bound and a good intuition for why gate counts are so punishing.
No — they mean the engineering targets are quantified. The whole point of error correction is that once physical fidelity clears the threshold, adding distance suppresses logical errors exponentially while costing only quadratically many qubits. The numbers here are the budget lines, and they are exactly the ones the field is engineering down.
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