Encoded Universality and Decoherence-Free Subspaces Control Seminar – Quantum Information Science and Technology UC Berkeley, Feb. 9, 2004 Julia Kempe CS Division and Dept. of Chemistry, University of California, Berkeley LRI, Universite de Paris-Sud, Paris, France

Towards nanotechnology Size of the components Number of components Speed Gordon Moore 1965 prevent or use quantum effects ? Theoretical limitations reached in 2020 !!! Apparition of quantum phenomena

Information is physical! Use the laws of quantum mechanics for the basic components of an information processing machine!  Quantum computing  Quantum cryptography  Quantum information  …

Main applications zCryptography yProtocol of unconditionally secure secret key distribution [Bennett, Brassard 84] Implementation : ~ 100 km zQuantum information yTeleportation [B, B, Crépeau, Jozsa, Peres, Wooters 93] Implementation [Bouwmeester, Pan, Mattle, Eibl, Weinfurter, Zeilinger 97] zAlgorithms yFactoring, discrete logarithm,... [Shor 94] yDatabase search [Grover 96] Num. of qubits ? 1995 : 2, 1998 : 3, 2002 : 8 [Chuang (IBM)] - 10 [Los Alamos]

Overview zBasic notions of quantum computing zStandard solutions: yUniversal gate set yQuantum Error Correcting Codes (QECC) zEncoded Universality zDecoherence-free Subsystems

The qubit Classical bit: b  {0,1} Probabilistic bit: probability distribution d  R + {0,1} such that || d || 1 =1.  d=(p,1-p) with p  [0,1] Quantum bit: |   C {0,1} such that || |  || 2 =1.  |  =  |0  +  |1  with |  | 2 + |  | 2 = 1 (Dirac notation)

Qubit evolution zMeasure: reads and modifies Measure | | 2| | 2 |  | 2|  | 2  |0  +  |1  |0|0 |1|1  Superposition  Probability distribution  Unitary transformation: U  C 2  2 such that UU † =Id |  U |  ’  = U |  unitary  reversible: U|  U†U† | 

Example Superposition: Measure: Measure 1/3 2/3 |0|0 |1|1 | 

Example Superposition: Measure: Unitary transformations: yNOT: |0   |1  yHadamard: Measure 1/3 2/3 |0|0 |1|1 |  U |  ’  = U |  H

Quantum computer: n qubits zn qubits  tensor product |   C {0,1} n such that || |  || 2 =1.  |  =  x  {0,1} n  x |x  with  x |  x | 2 = 1 zMeasure zPartial Measure Measure |x | 2|x | 2  x  {0,1} n  x |x  |x|x Measure Second bit = 0 (|  | 2 + |  | 2 )  |00  +  |01  +  |10  +  |11 

Quantum computer: n qubits  n qubits  tensor product |   C {0,1} n such that || |  || 2 =1.  |  =  x  {0,1} n  x |x  with  x |  x | 2 = 1 zMeasure zPartial Measure zUnitary transformation |   U|  with U  U(2 n ) ex: XOR= Measure |x | 2|x | 2  x  {0,1} n  x |x  |x|x Measure Second bit = 0 (|  | 2 + |  | 2 )  |00  +  |01  +  |10  +  |11  |00  |01  |10  |11  + |i  |j  |i  |XOR(i,j) 

Quantum computing a function Let f: {0,1} n  {0,1} m x  f(x) Reversible: R f :{0,1} n+m  {0,1} n+m (x,y)  (x,y  f(x)) Quantum: U f  U(2 n+m ): C n+m  C n+m |x  |y   |x  |y  f(x) 

Simplest Quantum Algorithm: Deutsch’s Problem Input: function f:{0,1}  {0,1} (in black box) Question: f constant (f(0)=f(1)) or balanced (f(0)  f(1)) ? Quantum black box (reversible): Algorithm: one query only!!! f |x|x |y|y |x|x |y  f(x)  f H H H |0|0 |1|1 Measure |0  -constant |1  -balanced

Simplest Quantum Algorithm: Deutsch’s Problem Input: function f:{0,1}  {0,1} (in black box) Question: f constant (f(0)=f(1)) or balanced (f(0)  f(1)) ? Quantum black box (reversible): Algorithm: one query only!!! f |x|x |y|y |x|x |y  f(x)  f H H H |0|0 |1|1 Measure |0  -constant |1  -balanced =0 if f balanced =0 if f constant

Where are we in practice? Several physical architectures have been proposed: zNMR, solid state, ion traps, superconducting qubits, optical cavities, photons zQubit = nuclear spin of atoms in a molecule (NMR), nuclear spin of (doped) atoms in a silicon donor (solid state), vibrational degrees of freedom of ions (ion trap), flux-degree of freedom (superconductors), polarisation (photons)

Five requirements for the implementation of quantum computation* 1.A scalable physical system with well characterized qubits 2.The ability to initialise the qubits to a simple state, such as |00…0  3.A “universal” set of quantum gates 4.Long relevant decoherence times, much longer than the gate operation time 5.A qubit-specific measurement capability * D. DiVincenzo, 97

Overview zBasic notions of quantum computing zStandard solutions: yUniversal gate set yQuantum Error Correcting Codes (QECC) zEncoded Universality zDecoherence-free Subsystems

Five requirements for the implementation of quantum computation* 1.A scalable physical system with well characterized qubits 2.The ability to initialise the qubits to a simple state, such as |00…0  3.A “universal” set of quantum gates 4.Long relevant decoherence times, much longer than the gate operation time 5.A qubit-specific measurement capability * D. DiVincenzo, 97

Universal computation Classical circuit model: Quantum circuit model: evaluates boolean functions can be constructed from universal local gates (ex.: NAND, COPY) 010…1010…1 bits    0000 unitary transformations U qubits |0  |1  |0  Measure U

Quantum circuits + + =U Barenco et al. ’95: Single-qubit gates and CNOT generate every unitary transformation!

Five requirements for the implementation of quantum computation* 1.A scalable physical system with well characterized qubits 2.The ability to initialise the qubits to a simple state, such as |00…0  3.A “universal” set of quantum gates 4.Long relevant decoherence times, much longer than the gate operation time 5.A qubit-specific measurement capability * D. DiVincenzo, 97

Main obstacle – decoherence! zQuantum information is very fragile zAny interaction with the environment disturbs the stored information zSolutions: quantum error correcting codes, decoherence-free subspaces, fault- tolerant computation …

Active noise protection: quantum error correction Quantum information is very fragile -how maintain quantum coherence? Quantum Error-Correcting Codes (QECCS)! Easiest classical code: repetition code: 0   111 Can correct one bitflip error (majority) 100,010,001  ,101,110  111

Active noise protection: quantum error correction Quantum information is very fragile -how maintain quantum coherence? Quantum Error-Correcting Codes (QECCS)! Easiest classical code: repetition code: 0   111 Can correct one bitflip error (majority) 100,010,001  ,101,110  111 Quantum: Quantum: many possible errors (bit flip, phase error, measurements…) “Quantum” repetition code? Impossible – no cloning principle.

Active noise protection: quantum error correction No cloning principle: No cloning principle: It is impossible to copy a quantum state.Proof: if then and By linearity:

Active noise protection: quantum error correction 3-qubit QECC: protects against phase errors: 3-qubit QECC: protects against bitflip errors: 9-qubit QECC: protection against all one qubit errors (Shor-code):

Overview zBasic notions of quantum computing zStandard solutions: yUniversal gate set yQuantum Error Correcting Codes (QECC) zEncoded Universality zDecoherence-free Subsystems

Quantum circuits + + =U Barenco et al. ’95: Single-qubit gates and CNOT generate every unitary transformation!

Hamiltonians=Interactions + + =U Unitaries are generated by Hamiltonians:

Hamiltonians=Interactions + + =U Single-qubit gates and CNOT generate every unitary transformation! Unitaries are generated by Hamiltonians: Hamiltonians describe the interactions (of qubits) in physical systems. Quantum engineers tweak the Hamiltonians to produce single qubit gates and CNOT.

Universality: The problem “Easy” and “hard” interactions (system-dependent) “Easy”: intrinsic interactions “natural” to the system, easy to tune, rapid “Hard”: slower, require higher device complexity, high decoherence System“Easy”“Hard” Photon- qubits Single qubit operations (linear optics) Two qubit operations (non-linearity, non-deterministic qc) Solid state Two qubit operations (ex. J-gates) Single qubit operations (ex. local focused B-fields) “Coherent- state” qubits Two qubit operations (beam-splitter) Single qubit operations (non-linearity, non-det.) Can we avoid “hard” interactions?

Almost every interaction is universal! Deutsch et al.(’95), Lloyd (‘95) : Almost any interaction on two qubits is universal. In the generic sense. Does not include the most frequent interactions. Nature is not generic! H ij H ji qubit iqubit jqubit iqubit j

Change of paradigm Traditionally: manipulate the physical system* to produce + + H 1,H 2,... * Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence...

Change of paradigm Traditionally: manipulate the physical system to produce + + Universal encoded computation Universal encoded computation: interactions given by the physical system find a way to make them universal HHH Encoding? H 1,H 2,...

Classical « Analogy » Two coins can only flip the two coins together « encode » « 0 »- « 1 » flip Encoded « coin »

Language of Hamiltonians Which interactions are universal? Given  =  H 1, H 2,…, H n  can one generate any unitary transformation (exactly or approximatively)? U(t) = exp(iHt)

Language of Hamiltonians Which interactions are universal? Given  =  H 1, H 2,…, H n  can one generate any unitary transformation (exactly or approximatively)? Possible compositions: scalar multiple 1) scalar multiple 2) linear combination 3) Lie bracket H has to generate Lie algebra su(N) of the unitary group SU(N)! U(t) = exp(iHt)

Lie Algebra of H Lie(H) closed under: 1) scalar multiplication 2) linear combination 3) Lie-bracket H has to generate Lie algebra su(N) of the unitary group SU(N)!

EX: Heisenberg interaction zomnipresent in solid state physics (« Easy ») zis not universal: had to be supplemented with single qubit gates On three qubits: E 12 E 13 E 23 (Pauli matrices)

The algebra L 3 (E) of E (3 qubits) the algebra L 3 (E) splits into irreducible components as: L 3 (E)  L 3 (E)  S 1  I 4  S 2  I 2 su(2)  S 2 Encoded qubit ?  su(2) 2 2 Lie algebra of E :

The algebra L 3 (E) of E (3 qubits) the algebra L 3 (E) splits into irreducible components as: L 3 (E)  L 3 (E)  S 1  I 4  S 2  I 2 su(2)  S 2 Encoded qubit ?  su(2) 2 2 Simulation of all operations of one qubit (su(2)) with L 3 (E) on the encoded qubit ! Lie algebra of E :

The algebra L n (E) of E (n qubits) the algebra L n (E) splits into irreducible components as: L n (E) ...

The algebra L n (E) of E (n qubits) the algebra L n (E) splits into irreducible components as: L n (E) ... Commutant L’ of L n (E) : L’ is generated by (« spin » algebra su(2)) As a Lie algebra L’ splits into irreducible representations of su(2).

Useful theorem... Universal computation “for free”? Let L be a †-closed algebra closed under multiplication and linear combination. Then the underlying space H is isomorphic to and L and its commutant L’ split as: where M(C d ) (M(C n )) is the algebra of all matrices on C d (C n ).

Useful theorem Let L be a †-closed algebra closed under multiplication and linear combination. Then the underlying space H is isomorphic to and L and its commutant L’ split as: where M(C d ) (M(C n )) is the algebra of all matrices on C d (C n ). The multiplicative algebra is not at our disposition! However the Lie algebra splits into irreducible components in the same basis: NO!

Problem of “Encoded Universality” Given an ensemble of generators H with Lie algebra Lie(H) which splits as can one find a component s.t. contains su(n j )? Encode the quantum information into the corresponding sub-space. dimension: n j... Yes

EX: The algebra L n (E) of E (n qubits) the algebra L n (E) splits into irreducible components as: L n (E) ... Results: Results: E is universal with encoding* introduce tensor structure, ex. blocks with 3 qubits** *J.K., D.Bacon, D.A.Lidar, K.B.Whaley, Phys. Rev. A 63: (2001) **D.DiVincenzo, D.Bacon, J.K., K.B.Whaley, NATURE 408 (2000) 19 operations for CNOT, 4 operations for 1-qubit

Ex: Heisenberg interaction On three qubits: E 12 E 13 E 23 (Pauli matrices)  su(2) 2 2 Proof idea: su(2) su(2)=Lie(  x,  y,  z ) [  x,  y ]= i  z [  y,  z ]= i  x [  z,  x ]= i  y [  i,  j ]= i  ijk  k

Overview zBasic notions of quantum computing zStandard solutions: yUniversal gate set yQuantum Error Correcting Codes (QECC) zEncoded Universality zDecoherence-free Subsystems

Decoherence-free subsystems (DFS) Decoherence-free subsystems (DFS) (“Dual” approach to encoded universality) zinteraction described by system-environment Hamiltonian: System (quantum computer) Environment interaction causes noise Decoherence system environmt.

DFS z Start initially decoupled from environment z Coupling terms in H I perturb the unitary evolution and evolve system into a mixed state system environmt.

DFS z Start initially decoupled from environment z Coupling terms in H I perturb the unitary evolution and evolve system into a mixed state z Solutions: Active (error correction) - QECC: deal well with independent errors on qubits Passive (error avoidance) – DFS: find a subspace of the system space over which evolution stays unitary, unperturbed, correlated noise noise-free subspace, exploits symmetries in the noise system environmt.

Classical « Analogy » (again!) Two coins Adversary (noise) can only flip the right coin encode a « noise-free » coin « 0 »- « 1 » flip Adversary (noise)  encoded 2 states of coin Manipulate encoded coin with global flips!

More formal the S  (and the algebra generated by them) split as: If we only keep the states within one component then we can encode into and the coupling H I will not affect the encoded states!... S S  encode into here system environmt.

eXAMPLE: Collective Decoherence splits into irreducible components as:... (« spin » algebra su(2))

eXAMPLE: Collective Decoherence splits into irreducible components as:... (« spin » algebra su(2)) DF- Subspaces 1-dim Start decoupled: unaffectd by H I Effective encoding: (=0)

eXAMPLE: Collective Decoherence splits into irreducible components as:... (« spin » algebra su(2)) DF- Subsystem* 2j+1-dim Each C 2j+1 unaffectd by H I *Knill, Laflamme, Viola, PRL’01

3 qubits 3 qubits splits as: S    S 4  I 1  S 2  I 2 2 H I doesn’t mix the two spaces! Commutant of the S  is Lie(E)!

Problem of DFS Given an interaction Hamiltonian where the S  split as find a component such that n j is large enough to encode quantum states ( “=“ DFS).

Problem of DFS Given an interaction Hamiltonian where the S  split as find a component such that n j is large enough to encode quantum states ( “=“ DFS). Moreover, in order to universally compute over the DFS, we need the commutant ! Computing with protected states  DF-state has to stay within DF-space during the entire operation  gate Hamiltonian has to commute with all the S 

Computing on DFS Computing with protected states  DF-state has to stay within DF-space during the entire operation  gate Hamiltonian has to commute with all the S ... S S 

Computing on DFS Computing with protected states  DF-state has to stay within DF-space during the entire operation  gate Hamiltonian has to commute with all the S ... S S  DF-Subspace:

Same as “Encoded Universality”! Given an ensemble of generators H with Lie algebra L(H) which splits as can one find a component s.t. contains su(n j )? Encode the quantum information into the corresponding sub-space. dimension: n j... Yes

Summary-Outlook DFS and Encoded Universality related concepts – representation theory of Lie algebras DFS: so far solved only for specific noise models – collective decoherence Encoded Universality: encodings so far for variants of the exchange interaction OPEN: other interactions/noise models hybrid DFS-QECC (so far concatentation)

References Encoded Universality: J. Kempe, D. Bacon, D.P. DiVincenzo, K.B. Whaley, “ Encoded universality from a single physical interaction”, in «Quantum Information and Computation»; Special Issue, Vol. 1, 2001, quant-ph/ D. Bacon, J. Kempe, D.P. DiVincenzo, D.A. Lidar, K.B. Whaley, “ Encoded Universality in Physical Implementations of a Quantum Computer”, Proceedings of IQC ’01, Australia, quant-ph/ D.P. DiVincenzo, D. Bacon, J. Kempe, K.B. Whaley, “ Universal Quantum Computation with the Exchange Interaction”, NATURE 408, 339 (2000), quant-ph/ J. Vala and K.B. Whaley, “Encoded Universality with Generalized Anisotropic Exchange Interactions”, Earlier related work on DFSs: D. Bacon, D. Lidar, K.B. Whaley, Phys. Rev. Lett. (2000) J. Kempe, D. Bacon, D. Lidar, K.B. Whaley, Phys. Rev. A 63: (2001) D. Bacon, J. Kempe, D. Lidar, K.B. Whaley, Phys. Rev. Lett. (2000)