Folklore on Qubits on Angus Lowe

Randomized Compilation

Thu, 26 Mar 2026 00:00:00 +0000

A growing body of work in the last few years points to randomization being a powerful tool for simulating quantum processes. For example, the fact that one can use a linear combination of noisy quantum channels to simulate the action of a unitary is the basis for the probabilistic error cancellation method in error mitigation. However, this is an example where noise is used to counteract noise, and the improvements gained are necessarily modest. Perhaps more strikingly, randomness can also sometimes give dramatically improved methods in the noise-free setting where one wishes to simulate unitaries with other unitaries. For instance, it has recently been shown¹ that an n-qubit Toffoli gate can be implemented with exponentially fewer T gates using randomly drawn circuits, and other works²^,³ draw attention to randomization reducing Trotter steps in Hamiltonian simulation. While we are used to thinking of uniquely quantum phenomena (e.g., entanglement) as resources for improving classical information processing tasks, these examples flip the script on this mantra: in the physically realistic setting of limited coherent control, classical randomness can be a resource for quantum information processing tasks.

Pauli Errors Suffice for Stabilizer Codes

Sat, 28 Feb 2026 00:00:00 +0000

Recall that a stabilizer is an Abelian subgroup of the $n$-qubit Pauli group

$$ \mathcal{G}_n = \{\pm 1,\pm i\}\cdot\{I,X,Y,Z\}^{\otimes n}. $$

We write $\mathcal{S}=\langle g_1,\dots, g_{n-k}\rangle$ to mean the stabilizer $\mathcal{S}$ is generated by the elements in angled brackets. If these elements are independent, the resulting subspace is $2^k$-dimensional, and hence can encode $k$ qubits. The code corresponding to $\mathcal{S}$ is the subspace $C\subseteq (\mathbb{C}^2)^{\otimes n}$ defined through

$$ C = \{\ket{\psi}\in(\mathbb{C}^2)^{\otimes n}: S\ket{\psi}=\ket{\psi}\ \forall S\in\mathcal{S}\}. $$

Clearly, we cannot have $-I\in \mathcal{S}$. This rules out phases of $\pm i$ appearing in the stabilizers.

Quantum Capacity of the Erasure Channel

Thu, 05 Feb 2026 00:00:00 +0000

This is a condensed version, with figures, of an argument appearing in Ref.¹

The quantum capacity $Q(p)$ of the erasure channel $\mathcal{N}:\rho\mapsto (1-p)\rho + p\lvert\perp\rangle\langle\perp\rvert$ is zero if $p\geq 1/2$ and

$$ Q(p)=1-2p \quad \forall p\in [0,1/2). $$

The attainability of this rate can be proven using random stabilizer codes and a probabilistic existence argument¹^,². (Perhaps I will update with this direction later.) In the remainder of this note we proof-sketch the upper bound.

Modern Quantum Error Correcting Conditions

Sun, 01 Feb 2026 00:00:00 +0000

We can explain the criteria for quantum error correction¹ starting from simple classical ideas and then promoting them to their quantum counterparts. This is my preferred rendition of the story because it contains a number of threads which you can follow to explore other fundamental results in information theory, quantum and otherwise. It is also a tale that aligns with modern teachings (in my experience), and makes explicit the connection to classical error correction.