Density Matrix Tomography, Contextuality, Future Spin Architectures T. S. Mahesh Indian Institute of Science Education and Research, Pune

1/2 Density Matrix Tomography (1-qubit)  = ~ MxMx MyMy PC = R+iS -P +   = ħ  / kT ~ Background Does not lead to signal Deviation May lead to signal

PC = R+iS -P Density Matrix Tomography (1-qubit) NMR detection operators:  x,  y 1. Heterodyne detection  x  = 2R  y  = -2S 2. Apply (  /2) y + Heterodyne detection  x  = 2P  = ~ MxMx MyMy (  /2) y - RP+iS R  1 =

P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I REAL NUMBERS Density Matrix Tomography (2-qubit) NMR detection operators:  x 1,  y 1,  x 2,  y 2

P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I REAL NUMBERS Traditional Method : Requires 1.Spin selective pulses 2.Integration of Transition Spin 1Spin 2 II 90 x I I 90 y I I 90 x 90 y 90 x 90 y Density Matrix Tomography (2-qubit)

P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I REAL NUMBERS Traditional Method : Spin 1Spin 2 II 90 x I I 90 y I I 90 x 90 y 90 x 90 y Requires 1.Spin selective pulses 2.Integration of Transition

P0 P1 P2 R1R2R3 R4R5 R6 + I1I2I3 I4I5 I REAL NUMBERS NEW Method Requires 1.No spin selective pulses 2.Integration of spins Density Matrix Tomography (2-qubit) JMR, 2010

Density Matrix Tomography (2-qubit) SVD  tomo

Density Matrix Tomography of singlet state Theory Expt RealImag Correlation = = 0.98 tr(  th   exp ) [tr(  th 2 ) tr(  exp 2 )] 1/2 JMR, 2010

Quantum Contextuality

Non- Contextuality 1. The result of the measurement of an operator A depends solely on A and on the system being measured. 2. If operators A and B commute, the result of a measurement of their product AB is the product of the results of separate measurements of A and of B. All classical systems are NON-CONTEXTUAL Physics Letters A (1990), 151,

 Measurement outcomes can be assigned, in principle, even before the measurement Non- Contextuality

Quantum Contextuality x2x2 x1x1 x1x2x1x2 z1z1 z2z2 z1z2z1z2 z1x2z1x2 x1z2x1z2 y1y2y1y  Measurement outcomes can not be pre-assigned even in principle N. D. Mermin. PRL 65, 3373 (1990). = 6 LHVT QM Eg. Two spin-1/2 particles PRL 101,210401(2008)

Laflamme, PRL 2010

~ 5.3 Laflamme PRL 2010 NMR demonstration of contextuality Sample: Malonic acid single crystal

Peres Contextuality Let us consider a system of two spin half particles in singlet state. Singlet state: Physics Letters A (1990), 151,

Peres Contextuality For a singlet state = -1 Note: [σ x 1,σ x 2 ] = 0 [σ y 1,σ y 2 ] = 0 [σ x 1 σ y 2, σ y 1 σ x 2 ] = 0 Physics Letters A (1990), 151,

Peres Contextuality For a singlet state Pre-assignment of eigenvalues = -1  x 1 x 2 = -1 = -1  y 1 y 2 = -1 = -1  x 1 y 2 y 1 x 2 = -1 CONTRADICTION !! Note: [σ x 1,σ x 2 ] = 0 [σ y 1,σ y 2 ] = 0 [σ x 1 σ y 2, σ y 1 σ x 2 ] = 0 Physics Letters A (1990), 151,

Experiment Using three F spins of Iodotrifluoroethylene. Two were used to prepare singlet and one was ancilla.

Pseudo-singlet state Pure singlet state is hard to prepare in NMR I z 1 +I z 2 +I z 3

Pseudo-singlet state Pure singlet state is hard to prepare in NMR I z 1 +I z 2 +I z 3 No Signal !! =0

Pseudo-singlet state Theory Experiment Real Part Imaginary Part Fidelity=0.97

Moussa Protocol Target (ρ) Probe(ancilla)|+  Target (ρ) Physical Review Letters (2010), 104, AB AB

NMR circuit for Moussa Protocol Singlet 1 (Ancilla) 2 3 B |+  A =

Results Manvendra Sharma, 2012

Future Architectures ?

Criteria for Physical Realization of QIP 1.Scalable physical system with mapping of qubits 2.A method to initialize the system 3.Big decoherence time to gate time 4.Sufficient control of the system via time-dependent Hamiltonians (availability of a universal set of gates). 5. Efficient measurement of qubits DiVincenzo, Phys. Rev. A 1998

NMR Circuits - Future Time Qubits  xx - qubits Decoherence Transverse relaxation  |00  +  |11  Loss of q. memory {|00 , |11  } Longitudinal relaxation |  |  Loss of c. memory T2T1 < Addressability Week coupling Controllability Larger Quantum register

Liquid-state NMR systems  Advantages  High resolution  Slow decoherence  Ease of control  Disadvantages o Smaller resonance dispersion o Small indirect (J) couplings o Smaller quantum register Random, isotropic tumbling

Single-crystal NMR systems  Advantages  Large dipole-dipole couplings ( > 100 times J)  Orientation dependent Hamiltonian  Longer longitudinal relaxation time  Larger quantum register (???)  Disadvantages o Shorter transverse relaxation time o Challenging to control the spin dynamics

Single-crystal NMR systems  Active spins in a bath of inactive molecules Large couplings High resolution Hopefully – Larger quantum register J. Baugh, PRA 2006

Two-molecules per unit center: Inversion symmetry – P1 space group So, the two molecules are magnetically equivalent Inter-molecular interactions ? Malonic Acid QIP with Single Crystals Cory et al, Phys. Rev. A 73, (2006)

Malonic Acid QIP with Single Crystals Cory et al, Phys. Rev. A 73, (2006) Natural Abundance

Pseudopure States Malonic Acid Cory et al, Phys. Rev. A 73, (2006)

Pseudopure States Malonic Acid Cory et al, Phys. Rev. A 73, (2006)

Quantum Gates Eg. C 2 -NOT Cory et al, Phys. Rev. A 73, (2006)

~ 5.3 R. Laflamme, PRL 2010

Glycine Single Crystal Mueller, JCP PPS

Floquet Register S. Ding, C. A. McDowell, … M. Liu, quant-ph/ More qubits More coupled Nuclear Spins More Resolved Transitions Side-bands?

S. Ding, C. A. McDowell, … M. Liu, quant-ph/

Solid-State NMR and next generation QIP Pseudo-Pure States 13 C spectra of aromatic carbons of Hexamethylbenzene spinning at 3.5 kHz

Grover’s Algorithm S. Ding, C. A. McDowell, … M. Liu, quant-ph/ Methyl 13 C

Electron Spin vs Nuclear Spin Spin e n Magnetic moment Sensitivity High Low Coherence Time Measurement Processing

e-n Entanglement Mehring, 2004 Entanglement in a solid-state spin ensemble Stephanie Simmons et alStephanie Simmons Nature 2011

Electron spin actuators Cory et al

Detection of single Electron Spin D. Rugar, R. Budakian, H. J. Mamin & B. W. Chui Nature 329, 430 (2004) by Magnetic Resonance Force Microscopy

 eq =  e  e   I  N U p = SWAP (e,n 1 ) I e   1  1   I  (N-1) Measure e-spin If  e  invert U p = SWAP (e,n 2 )  e  e    1  1   I  (N-1) Cooling of nuclear spins Cory et al, PRA 07

Nuclear Local Fields under Anisotropic Hyperfine Interaction B0B0 e-n system

Coherent oscillations between nuclear coherence on levels 1 & 2 driven by Microwave The nuclear  pulse : 520 ns e-n CNOT gate : 2  s (0.98 Fidelity) Anisotropic Hyperfine Interaction