Qudit circuits with SU(d) symmetry: Locality imposes additional conservation laws

Local symmetric quantum circuits provide a simple framework to study the dynamics and phases of complex quantum systems with conserved charges. However, some of their basic properties have not yet been understood. Recently, it has been shown that such quantum circuits only generate a restricted subset of symmetric unitary transformations [I. Marvian, Nature Physics, 2022]. In this paper, we consider circuits with 2-local SU(d)invariant unitaries acting on qudits, i.e., d-dimensional quantum systems. Our results reveal a significant distinction between the cases of d = 2 and d ≥ 3. For qubits with SU(2) symmetry, arbitrary global rotationallyinvariant unitaries can be generated with 2-local ones, up to relative phases between the subspaces corresponding to inequivalent irreducible representations (irreps) of the symmetry, i.e., sectors with different angular momenta. On the other hand, for d ≥ 3, in addition to similar constraints on the relative phases between the irreps, locality also restricts the generated unitaries inside these conserved subspaces. These constraints impose conservation laws that hold for dynamics under 2-local SU(d)-invariant unitaries, but are violated under general SU(d)invariant unitaries. Based on this result, we show that the distribution of unitaries generated by random 2-local SU(d)-invariant unitaries does not converge to the Haar measure over the group of all SU(d)-invariant unitaries, and in fact, for d ≥ 3, is not even a 2-design for the Haar distribution.

[1]  S. Sondhi,et al.  Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws , 2017, 1705.08910.

[2]  A. Harrow,et al.  Random Quantum Circuits are Approximate 2-designs , 2008, 0802.1919.

[3]  J. Renes,et al.  Beyond heat baths: Generalized resource theories for small-scale thermodynamics. , 2014, Physical review. E.

[4]  David Jennings,et al.  Description of quantum coherence in thermodynamic processes requires constraints beyond free energy , 2014, Nature Communications.

[5]  Xiao-Gang Wen,et al.  Classification of gapped symmetric phases in one-dimensional spin systems , 2010, 1008.3745.

[6]  A. Harrow.  Applications of coherent classical communication and the schur transform to quantum information theory , 2005, quant-ph/0512255.

[7]  G. Gour,et al.  Low-temperature thermodynamics with quantum coherence , 2014, Nature Communications.

[8]  Barry C. Sanders,et al.  Entangling power and operator entanglement in qudit systems , 2003 .

[9]  Siddhartha Santra,et al.  Ensembles of physical states and random quantum circuits on graphs , 2012 .

[10]  DiVincenzo.  Two-bit gates are universal for quantum computation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[11]  D. Janzing,et al.  Thermodynamic Cost of Reliability and Low Temperatures: Tightening Landauer's Principle and the Second Law , 2000, quant-ph/0002048.

[12]  R. Zeier,et al.  On squares of representations of compact Lie algebras , 2015, 1504.07732.

[13]  Nicholas J. Mayhall,et al.  Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm , 2019, npj Quantum Information.

[14]  F. Brandão,et al.  Local random quantum circuits are approximate polynomial-designs: numerical results , 2012, 1208.0692.

[15]  R. Spekkens,et al.  The resource theory of quantum reference frames: manipulations and monotones , 2007, 0711.0043.

[16]  Jeongwan Haah,et al.  Quantum Entanglement Growth Under Random Unitary Dynamics , 2016, 1608.06950.

[17]  B. Sanders,et al.  Quantum encodings in spin systems and harmonic oscillators , 2001, quant-ph/0109066.

[18]  Domenico D'Alessandro,et al.  Uniform finite generation of compact Lie groups , 2002, Syst. Control. Lett..

[19]  J. Maldacena,et al.  A bound on chaos , 2015, Journal of High Energy Physics.

[20]  T. Rudolph,et al.  A relational quantum computer using only two-qubit total spin measurement and an initial supply of highly mixed single-qubit states , 2005, quant-ph/0503151.

[21]  F. Brandão,et al.  Resource theory of quantum states out of thermal equilibrium. , 2011, Physical review letters.

[22]  T. Matsubara,et al.  A Lattice Model of Liquid Helium , 1956 .

[23]  D. Markham,et al.  Quantum secret sharing with qudit graph states , 2010, 1004.4619.

[24]  L'algèbre de Lie des transpositions , 2005, math/0502119.

[25]  Jeongwan Haah,et al.  Operator Spreading in Random Unitary Circuits , 2017, 1705.08975.

[26]  A. J. Short,et al.  Thermodynamics of quantum systems with multiple conserved quantities , 2015, Nature Communications.

[27]  T. Rudolph,et al.  Quantum coherence, time-translation symmetry and thermodynamics , 2014, 1410.4572.

[28]  L. Susskind,et al.  Complexity and Shock Wave Geometries , 2014, 1406.2678.

[29]  Seth Lloyd,et al.  Convergence conditions for random quantum circuits , 2005, quant-ph/0503210.

[30]  K. B. Whaley,et al.  Theory of decoherence-free fault-tolerant universal quantum computation , 2000, quant-ph/0004064.

[31]  Fisher,et al.  Boson localization and the superfluid-insulator transition. , 1989, Physical review. B, Condensed matter.

[32]  Fernando G S L Brandão,et al.  Efficient Quantum Pseudorandomness. , 2016, Physical review letters.

[33]  M. Horodecki,et al.  Fundamental limitations for quantum and nanoscale thermodynamics , 2011, Nature Communications.

[34]  Siddhartha Santra,et al.  Quantum entanglement in random physical states. , 2011, Physical review letters.

[35]  Daniel Burgarth,et al.  Symmetry criteria for quantum simulability of effective interactions , 2015, 1504.07734.

[36]  Amir Kalev,et al.  Noncommuting conserved charges in quantum many-body thermalization. , 2020, Physical review. E.

[37]  R. Spekkens,et al.  Extending Noether’s theorem by quantifying the asymmetry of quantum states , 2014, Nature Communications.

[38]  A. Ekert,et al.  Universality in quantum computation , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[39]  Linghang Kong,et al.  Charge-conserving unitaries typically generate optimal covariant quantum error-correcting codes , 2021, 2102.11835.

[40]  D. Huse,et al.  Operator Spreading and the Emergence of Dissipative Hydrodynamics under Unitary Evolution with Conservation Laws , 2017, Physical Review X.

[41]  Ivan Marin.  Group Algebras of Finite Groups as Lie Algebras , 2008, 0809.0074.

[42]  T. Rudolph,et al.  Reference frames, superselection rules, and quantum information , 2006, quant-ph/0610030.

[43]  Robert W. Spekkens,et al.  Asymmetry properties of pure quantum states , 2012 .

[44]  Lloyd,et al.  Almost any quantum logic gate is universal. , 1995, Physical review letters.

[45]  L. Susskind.  Computational complexity and black hole horizons , 2014, 1402.5674.

[46]  F. Pollmann,et al.  Diffusive Hydrodynamics of Out-of-Time-Ordered Correlators with Charge Conservation , 2017, Physical Review X.

[47]  David Jennings,et al.  Thermodynamic resource theories, non-commutativity and maximum entropy principles , 2015, 1511.04420.

[48]  P. Zanardi,et al.  Virtual quantum subsystems. , 2001, Physical review letters.

[49]  J. Levy.  Universal quantum computation with spin-1/2 pairs and Heisenberg exchange. , 2001, Physical review letters.

[50]  A. Winter,et al.  Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges , 2015, Nature Communications.

[51]  S. Shenker,et al.  Black holes and the butterfly effect , 2013, Journal of High Energy Physics.

[52]  I. Marvian,et al.  Building all time evolutions with rotationally invariant Hamiltonians , 2008, 0802.0870.

[53]  Daniel A. Roberts,et al.  Chaos and complexity by design , 2016, 1610.04903.

[54]  Daniel A. Roberts,et al.  Holographic Complexity Equals Bulk Action? , 2016, Physical review letters.

[55]  K. B. Whaley,et al.  Universal quantum computation with the exchange interaction , 2000, Nature.