Iman Marvian

According to a fundamental result in quantum computing, any unitary transformation on a composite system can be generated using so-called 2-local unitaries that act only on two subsystems. Beyond its importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time.  In a recent paper, I have recently shown that this universality does not hold in the presence of conservation laws and global continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems. Based on this no-go theorem,  I propose a method for experimentally probing the locality of interactions in nature. In the context of quantum thermodynamics, our results mean that generic energy-conserving unitary transformations on a composite system cannot be realized solely by combining local energy-conserving unitaries on the components. I show how this can be circumvented via catalysis.

In two follow-up papers (here and here)  with Hanqing Liu and Austin Hulse, we study the case of SU(d) symmetry for qudits.   We show that in the case of d=2, i.e., qubits with SU(2) symmetry, any rotationally-invariant unitary on qubits can be realized using the Heisenberg exchange interaction, which is 2-local and rotationally-invariant, provided that the qubits in the system interact with a pair of ancilla qubits.  Our results reveal a significant distinction between the cases of d = 2 and d>2. For qubits with SU(2) symmetry, arbitrary global rotationally-invariant 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>2, 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>2, is not even a 2-design for the Haar distribution.

Furthermore, we show that any rotationally-invariant unitary on qubits can be realized using Heisenberg interaction, provided that the system can interact with a pair of ancilla qubits.

Related papers

If one looks at the scientific history of the theory of entanglement, the turning point is easily seen to occur in the mid-nineties, at the point when researchers in quantum information theory began to consider entanglement as “a resource as real as energy”. It gradually became clear that the entanglement theory should be understood as a framework to study questions about manipulating resource states for performing certain tasks, similar to the theory of thermodynamics. From this point on, entangled states and entangling operations were defined as those states and operations that cannot be implemented when one only has access to Local Operations and Classical Communication. Researchers then began to systematically answer questions such as: under this kind of restriction when is it possible to convert one resource state into another? How do we quantify the resource? What is the resource cost of simulating an operation? Subsequently, motivated by the success of the resource theory approach to entanglement, many researchers started applying this approach to understand other properties of quantum systems, such as coherence, asymmetry and athermality in quantum thermodynamics.

1. I. Marvian, Operational Interpretation of Quantum Fisher Information in Quantum Thermodynamics, To appear in Phys. Rev. Lett (2022), arXiv preprint arXiv:2112.04694. 
2. I. Marvian,  Coherence distillation machines are impossible in quantum thermodynamics, Nature Communications (2020).
3. I. Marvian, R.W. Spekkens, No-Broadcasting Theorem for Quantum Asymmetry and Coherence and a Trade-off Relation for Approximate Broadcasting, Phys. Rev. Lett. 123, 020404 (2019).
4. G. Gour, D. Jennings, F. Buscemi, R. Duan, I. Marvian, Quantum majorization and a complete set of entropic conditions for quantum thermodynamics,  Nature Communications.
5. I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94 , 052324  (2016 ), Editors’ Suggestion.
6. I. Marvian and R. W. Spekkens, Extending Noether’s theorem by quantifying the asymmetry of quantum states, Nature Communications 5 , 3821  (2014 ), arXiv:1404.3236.
7. A. Kolchinsky, I. Marvian, C. Gokler,  Z. Liu, , P. Shor, O. Shtanko, K. Thompson, D.Wolpert, S. Lloyd, Maximizing free energy gain, arXiv:1705.00041.
8. I. Marvian and R. W. Spekkens, The theory of manipulations of pure state asymmetry I: basic tools and equivalence classes of states under symmetric operations, New J. Phys. 15, 033001  (2013).
9. I. Marvian and R. W. Spekkens, Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames, Phys. Rev. A 90, 062110  (2014 ).

1. A Gilyén, S Lloyd, I Marvian, Y Quek, MM Wilde, Quantum algorithm for Petz recovery channels and pretty good measurements, Physical Review Letters 128 (22), 220502 (2022).
2. M. Kjaergaard, et al., Demonstration of density matrix exponentiation using a superconducting quantum processor, Physical Review X 12 (1), 011005  (2022).
3. A. Steffens, P. Rebentrost, I. Marvian, J. Eisert, S. Lloyd, An efficient quantum algorithm for spectral estimation, New J. Phys. 19  (3 ), 033005  (2017 ), arXiv:1609.08170 .
4. P. Rebentrost, A Steffens, I. Marvian, S. Lloyd, Quantum singular-value decomposition of nonsparse low-rank matrices, Physical Review A 97 (1), 012327 (2018).

5. I. Marvian and S. Lloyd, Universal Quantum Emulator, arXiv:1606.02734.

1. I. Marvian and D. A. Lidar, Quantum speed limits for leakage and decoherence, Phys. Rev. Lett. 115 , 210402  (2015 ), arXiv:1505.07850 .
2. I. Marvian and D. A. Lidar, Quantum error suppression with commuting Hamiltonians: Two-local is too local , Phys. Rev. Lett. 113 , 260504  (2014 ), arXiv:1410.5487. 

3. I. Marvian, Exponential suppression of decoherence and relaxation of quantum systems using energy penalty, arXiv:1602.03251.

1. I. Marvian, Symmetry-Protected Topological Entanglement, Phys. Rev. B 95 , 045111  (2017 ), arXiv:1307.6617.

1. I. Marvian, R.W. Spekkens, and Paolo Zanardi, Quantum speed limits, coherence and asymmetry, Phys. Rev. A 93 , 052331  (2016 ), Editors’ Suggestion, arXiv:1510.06474.

2. I. Marvian and D. A. Lidar, Quantum speed limits for leakage and decoherence, Phys. Rev. Lett. 115 , 210402  (2015 ), arXiv:1505.07850 .

3. P. Coles, V. Katariya, S. Lloyd, I. Marvian, M. Wilde, Entropic Energy-Time Uncertainty Relation, arXiv:1805.07772.