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Logical Qubits vs Physical Qubits: Why 1 Million Qubits Isn't What You Think

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Two Numbers, Very Different Meanings

When a headline says "PsiQuantum is building 1 million qubits" or "IBM Condor has 1,121 qubits", the number in question is almost always physical qubits. This is the count of actual quantum hardware elements. For cryptanalysis, the number that matters is logical qubits, the error-corrected, protocol-usable units.

The ratio between them is enormous. Current surface code implementations need roughly 1,000 physical qubits to produce a single logical qubit of useful quality.

Physical Qubits

A physical qubit is the raw hardware. Depending on technology:

Superconducting: a Josephson junction chilled to ~10 millikelvin.
Trapped ion: a single charged atom suspended in an electromagnetic trap.
Photonic: a single photon routed through a waveguide.
Neutral atom: a single neutral atom held in an optical tweezer.

Physical qubits have finite coherence times (microseconds to seconds depending on technology) and finite gate fidelities (typically 10^-3 to 10^-4 error per two-qubit gate in 2026).

Logical Qubits

A logical qubit is an abstraction built on many physical qubits through a quantum error correcting code. The most common code in practice is the surface code. It represents one logical qubit as a d-by-d grid of physical qubits, where d is the "code distance". A larger d yields lower error but higher physical-qubit cost.

For code distance d, the logical error rate decreases exponentially with d as long as the underlying physical error rate is below the code's threshold (roughly 1 percent for the surface code).

Typical costs:

Code distance d	Physical qubits per logical	Logical error rate (approx)
3	~17	10^-3
5	~49	10^-5
7	~97	10^-7
11	~241	10^-11
25	~1,249	10^-23

Google Willow (December 2024) demonstrated that scaling from d=3 to d=5 to d=7 halved the logical error rate at each step, which is the expected behavior once the hardware is below threshold. This was the real significance of Willow, not the 105 physical qubit count.

The Numbers for Breaking RSA

Gidney and Ekera's 2021 analysis (arXiv:1905.09749) estimates approximately 20 million noisy physical qubits at an error rate of 10^-3 to factor RSA-2048 in 8 hours. That breaks down to roughly 4,000 logical qubits at code distance ~25, with the rest of the physical qubits devoted to magic state factories and routing.

Some analyses come in lower (a few million physical qubits with more efficient codes and encoding tricks), others higher. No analysis puts the requirement inside the current hardware envelope.

What This Means Practically

A press release about "1 million qubits" almost always means 1 million physical. Under surface code with d=25, that gives you ~800 logical qubits. Still short of the ~4,000 needed for RSA-2048.
The scaling is not a straight doubling. Halving the physical qubit error rate is worth more than doubling the physical qubit count, because logical error drops exponentially in d.
PsiQuantum's stated goal of 1 million photonic qubits is a company projection, not a validated scientific timeline. It would still require a surface code distance that yields thousands of logical qubits to be cryptographically relevant.

Why Security Teams Should Care

If you are tracking the CRQC timeline to decide when to migrate, the metric is logical qubits, not physical. A 10,000-qubit superconducting processor at today's error rates would support roughly 10 logical qubits at d=25 code distance. That is still zero progress on RSA.

The milestones to watch are:

Logical qubit count past 100. Meaningful progress toward cryptanalysis.
Logical error rate below 10^-10. Required for Shor on 2048-bit integers.
Magic state factory throughput. Often the real bottleneck in Shor implementations.

Until then, qubit-count headlines are not a CRQC signal. They are a scaling signal. Scaling without reducing error rates is empty.

Frequently Asked Questions

How many physical qubits per logical qubit in practice?

Depends on code distance and target logical error rate. Typical target is 10^-10, requiring code distance ~25 in surface codes, which is roughly 1,250 physical qubits per logical qubit. Lower error rates, higher multiplier.

Did Google Willow produce a logical qubit?

Yes. Willow demonstrated a surface code at distances 3, 5, and 7. The important result was that the logical error rate dropped exponentially as code distance increased, confirming operation below the surface code threshold. It is one logical qubit, not enough for cryptanalysis, but a scientific milestone.

Is there a better code than the surface code?

Research continues. Color codes, LDPC codes, and concatenated codes each have tradeoffs. For near-term hardware, the surface code has the best tolerance for the 2D connectivity of superconducting chips.

Why not just build enough physical qubits to ignore error correction?

Because the error accumulates. Running Shor's algorithm requires billions of gate operations. At a per-gate error rate of 10^-3, you would accumulate errors long before completing the calculation. Error correction is not optional.

Sources

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Originally published at quantumsequrity.com/blog/logical-qubits-vs-physical-qubits.