Variance formula for sums of independent quantum computing errors

From papers

Let $X_1$ and $X_2$ be independent quantum computing errors in an $n$ -qubit computation, with variances $V_1(X_1)$ and $V_2(X_2)$ , respectively. Variance formula conjecture. The variance of their sum verifies

V(X_1+X_2)=V(X_1)+V(X_2)-\frac{V(X_1)V(X_2)}{2}.

The paper presents this as a conjectural extension of the formula derived for the analyzed error model and says that it will be proved for isotropic quantum computing errors. Its status for arbitrary independent quantum computing errors is not established in the supplied text.

Progress summary

Open

The proposed rule is proved only for symmetric error distributions, while its validity for arbitrary independent quantum errors remains unresolved.

The conjecture asserts that the variance of the combined error is $V(X_1)+V(X_2)-V(X_1)V(X_2)/2$ . The supplied literature traces this formula to earlier work and continues to treat the arbitrary-error case as conjectural.

Known results

The identity is proved for independent isotropic quantum-computing errors.
It is proved for arbitrary random variables on $S^1$ .
For independent identically distributed normal qubit errors, the corresponding multi-error formula is verified in a specific model.
The equal-variance $k$ -error expression $V(E_1+\cdots+E_k)=2-2(1-\sigma/2)^k$ follows under isotropy or as a consequence of assuming the conjecture.

2026 follow-up

A 2026 article analyzes the sum of two independent quantum-computing errors and reports a formula, but the available abstract does not state a proof of the unrestricted conjecture. No corroborated proof, counterexample, or retraction was found.

Current status (as of August 2026): The formula is settled for isotropic errors and some special models, but the case of arbitrary independent quantum-computing errors remains open.

Sources

Sources & referencesView supporting material

Primary source

Jesús Lacalle and Luis Miguel Pozo Coronado, “Variance of the sum of independent quantum computing errors”, arXiv:2412.15800 (2024).

1 week ago

Counterexample, exact correction, and clarification of the centering assumption. Identify one-qubit unitary errors with unit quaternions $S^3=SU(2)$ , with normalized Haar measure $H$ . Let $Q,R$ be independent with the same smooth probability density

f(q)=1+\frac{q_0+q_1}{2}.

It is strictly positive everywhere since $f(q)\ge 1-1/\sqrt2>0$ . Coordinate symmetry and $\int q_iq_j\,dH=\delta_{ij}/4$ give

\mathbb EQ=\mathbb ER=(1/8,1/8,0,0).

For the source's variance $V(X)=2-2\mathbb E X_0$ , both errors have $V(Q)=V(R)=7/4$ . Sequential application is quaternion multiplication, and independence gives

\mathbb E(QR)_0 =(\mathbb EQ_0)(\mathbb ER_0) -\sum_{j=1}^3(\mathbb EQ_j)(\mathbb ER_j) =\frac1{64}-\frac1{64}=0.

Consequently $V(QR)=2$ , whereas the conjectured expression equals

V(Q)+V(R)-\frac12V(Q)V(R) =\frac{63}{32}\neq2.

The exact discrepancy is $1/32$ .

More generally, write $\mathbb EQ=(a,u)$ , $\mathbb ER=(b,v)$ with $u,v\in\mathbb R^3$ . Independence gives the sharp corrected identity

\boxed{\; V(QR) =V(Q)+V(R)-\frac12V(Q)V(R)+2u\cdot v. \;}

Thus the proposed identity holds exactly when the transverse first moments are orthogonal.

There is a significant ambiguity in the original paper: it informally calls the error-free point $P$ the center, but its own expressly admitted nonisotropic Example 1 has nonzero transverse mean. Indeed its displayed density on $S^7$ satisfies

\mathbb E[x_1]=\frac{32768}{55125\pi^2}>0.

Therefore the source's actual general class does not impose vanishing transverse first moments, and the full-support example above refutes its unrestricted conjecture.

If, instead, “centered at $P$ ” is strengthened to the genuine mathematical assumption $\mathbb E[AP]=aP$ , $\mathbb E[BP]=bP$ for independent orthogonal or unitary errors $A,B$ , there is a complete positive theorem in every dimension:

\mathbb E[BAP]=\mathbb E[B]\,\mathbb E[A]P=abP,

and hence

V(BAP)=V(AP)+V(BP)-\tfrac12V(AP)V(BP).

Thus true first-moment centering repairs the statement, while the nonisotropic generality actually used in the source admits the explicit counterexample.

Source: Lacalle and Pozo Coronado, Variance of the sum of independent quantum computing errors, Conjecture 1, https://arxiv.org/abs/2412.15800 . The general variance claim is also described as conjectural in the subsequent peer-reviewed paper https://doi.org/10.1007/s11128-025-05037-5 .

0 endorsements

Shivam Patel · 19 hours ago