Variance formula for sums of independent quantum computing errors

At least 1 year old · documented by 2024

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written calculation claims the rule is false for general errors, but this counterexample has not been independently verified.

Lacalle and Pozo Coronado proposed the formula for sums of independent quantum-computing errors in December 2024, proving it for isotropic errors while leaving the unrestricted case conjectural: $V(X_1+X_2)=V(X_1)+V(X_2)-V(X_1)V(X_2)/2$ .

Known results

Independent isotropic errors in $S^d$ (Lacalle and Pozo Coronado, 2024).
Arbitrary independent errors on $S^1$ (Lacalle and Pozo Coronado, 2024).
Specific nonisotropic examples, including one on $S^7$ , satisfy the formula but do not establish the general case (2024).
A 2026 paper studies fidelity of sums but its available abstract does not claim a proof of this variance conjecture.

Posted attempt

A reader-written quaternion calculation claims a full-support one-qubit counterexample: it gives $V(Q)=V(R)=7/4$ , $V(QR)=2$ , while the conjectured value is $63/32$ , and proposes a correction term $2u\cdot v$ . The attempt has not been independently verified.

Current status (as of August 2026): The formula is proved for isotropic errors and certain special cases, while the arbitrary independent case is now challenged by an unverified counterexample and is not settled.

Sources

CounterexampleThis solution needs a summarySee full solutionHide full solution

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 .