Rotationally-Invariant Circuits: Universality with the exchange interaction and two ancilla qubits

Abstract

Universality of local unitary transformations is one of the cornerstones of quantum computing with many applications and implications that go beyond this field. However, it has been recently shown that this universality does not hold in the presence of continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems [I. Marvian, Nature Physics (2022)]. In this work, we study qubit circuits formed from k-local rotationally-invariant unitaries and fully characterize the constraints imposed by locality on the realizable unitaries. We also present an interpretation of these constraints in terms of the average energy of states with a fixed angular momentum. Interestingly, despite these constraints, we show that, using a pair of ancilla qubits, any rotationally-invariant unitary can be realized with the Heisenberg exchange interaction, which is 2-local and rotationally-invariant. We also show that a single ancilla is not enough to achieve universality. Finally, we discuss applications of these results for quantum computing with semiconductor quantum dots, quantum reference frames, and resource theories.

arXiv comment5 pages + 28 pages of Supplementary Material (The preliminary version). See also the related work on SU(d) symmetry in arXiv:2105.12877

References in corpus (46)

• Double-slit photoelectron interference in strong-field ionization of the neon dimer
• Quantum Computation with Quantum Dots
• Decoherence Free Subspaces for Quantum Computation
• Quantum principal component analysis
• Quantum Resource Theories
• Two-Bit Gates are Universal for Quantum Computation
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Universal Quantum Computation with the Exchange Interaction
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Reference frames, superselection rules, and quantum information
• Fundamental limitations for quantum and nano thermodynamics
• The Resource Theory of Quantum States Out of Thermal Equilibrium
• Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation
• Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
• Quantum coherence, time-translation symmetry and thermodynamics
• The resource theory of quantum reference frames: manipulations and monotones
• Universal quantum computation with spin-1/2 pairs and Heisenberg exchange
• Universal Fault-Tolerant Computation on Decoherence-Free Subspaces
• Universality in Quantum Computation
• Virtual Quantum Subsystems
• The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations
• Dynamical Generation of Noiseless Quantum Subsystems
• Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges
• Suppression of photon shot noise dephasing in a tunable coupling superconducting qubit
• Thermodynamics of quantum systems with multiple conserved quantities
• Thermodynamic resource theories, non-commutativity and maximum entropy principles
• Measurement Protocol for the Entanglement Spectrum of Cold Atoms
• Continuous symmetries and approximate quantum error correction
• Restrictions on realizable unitary operations imposed by symmetry and locality
• Coherence-Preserving Quantum Bits
• Noncommuting conserved quantities in quantum many-body thermalization
• Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ansätze
• Critical phase and spin sharpening in SU(2)-symmetric monitored quantum circuits
• Noncommuting conserved charges in quantum thermodynamics and beyond
• Non-Abelian symmetry can increase entanglement entropy
• Experimental observation of thermalization with noncommuting charges
• Error Correction of Quantum Reference Frame Information
• Symmetry criteria for quantum simulability of effective interactions
• Near-optimal covariant quantum error-correcting codes from random unitaries with symmetries
• Optimal Universal Quantum Error Correction via Bounded Reference Frames
• Universal quantum control in irreducible state-space sectors: application to bosonic and spin-boson systems
• Controllability of Symmetric Spin Networks
• Building all Time Evolutions with Rotationally Invariant Hamiltonians
• Universal control of quantum subspaces and subsystems
• A relational quantum computer using only two-qubit total spin measurement and an initial supply of highly mixed single qubit states
• Units of rotational information

Cited by in corpus (10)

• Theory of Quantum Circuits with Abelian Symmetries
• Designs from Local Random Quantum Circuits with SU(d) Symmetry
• Unitary Designs of Symmetric Local Random Circuits
• Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU($d$) Symmetry
• Characterization of randomness in quantum circuits of continuous gate sets
• Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras
• Architectures and random properties of symplectic quantum circuits
• Entanglement dynamics and Page curves in random permutation circuits
• SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning
• Non-Universality from Conserved Superoperators in Unitary Circuits