Experimental Implementation of Efficient Quantum Pseudorandomness on a 12-Spin System.

Quantum pseudorandomness, also known as unitary designs, comprises a powerful resource for emergent quantum technologies. Although in theory pseudorandom unitary operators can be constructed efficiently, realizing these objects in realistic physical systems is a challenging task. Here, we demonstrate experimental generation and detection of quantum pseudorandomness on a 12-qubit nuclear magnetic resonance system.

Private quantum subsystems.

We investigate the most general notion of a private quantum code, which involves the encoding of qubits into quantum subsystems and subspaces. We contribute to the structure theory for private quantum codes by deriving testable conditions for private quantum subsystems in terms of Kraus operators for channels, establishing an analogue of the Knill-Laflamme conditions in this setting. For a large class of naturally arising quantum channels, we show that private subsystems can exist even in the absence of private subspaces.

Generalization of quantum error correction via the Heisenberg picture.

We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called "operator algebra quantum error correction"). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems.

Method to find quantum noiseless subsystems.

We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operations.

Unified and generalized approach to quantum error correction.

We present a unified approach to quantum error correction, called operator quantum error correction. Our scheme relies on a generalized notion of a noiseless subsystem that is investigated here. By combining the active error correction with this generalized noiseless subsystems method, we arrive at a unified approach which incorporates the known techniques--i.