Experimental Quantum Simulation of Entanglement in Many-body Systems
aa r X i v : . [ qu a n t - ph ] J u l Experimental Quantum Simulation of Entanglement in Many-body Systems
Jingfu Zhang, Tzu-Chieh Wei , , and Raymond Laflamme , Institute for Quantum Computing and Department of Physics,University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2J 2W9, Canada Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada (Dated: November 2, 2018)We employ a nuclear magnetic resonance (NMR) quantum information processor to simulatethe ground state of an XXZ spin chain and measure its NMR analog of entanglement, or pseudo-entanglement. The observed pseudo-entanglement for a small-size system already displays singular-ity, a signature which is qualitatively similar to that in the thermodynamical limit across quantumphase transitions, including an infinite-order critical point. The experimental results illustrate asuccessful approach to investigate quantum correlations in many-body systems using quantum sim-ulators.
PACS numbers: 03.67.Ac, 03.67.Lx, 75.10.Pq
Entanglement, delineated as the non-local correlation,is one “spooky” characteristic trait of quantum mechan-ics [1]. The famous dispute between Bohr and Ein-stein on the fundamental question of quantum mechanics,Schr¨odinger-cat paradox, and the transitions from quan-tum to classical worlds essentially involve entanglement.Recent development of quantum information rekindlesthe interest in entanglement, more as a possible resourcefor information processing [2]. Various methods havebeen proposed to characterize entanglement qualitativelyand quantitatively [3]. One immediate application of theentanglement is the investigation of quantum phase tran-sitions (QPTs) [4–6] in many-body systems, which oc-cur at zero temperature (T=0 K), where the transitionsare driven by quantum fluctuations and the ground-statewave function is expected to develop drastic change. Theentanglement properties extend and complement the tra-ditional statistical-physical methods for QPTs, such asthe correlation functions and low-lying excitation spec-tra. However, how to describe and measure entanglementin many-body systems is still a challenging task in boththeoretical and experimental aspects [7, 8]. Most schemesfor directly measuring entanglement focus on the entan-glement between two qubits [9, 10]. Although the degreeof entanglement for a medium-size or lager system can inprinciple be probed [8, 11], it has not been experimentallymeasured directly.In contrast to classical approaches, quantum simula-tors [12] provide a promising approach for investigatingmany-body systems and enable one to efficiently simu-late other quantum systems by actively controlling andmanipulating a certain quantum system, and to test,probe and unveil new physical phenomena. One inter-esting aspect is to simulate the ground states of many-body systems, where usually rich phases can exist, suchas ferromagnetism, superfluidity, and quantum Hall ef-fect, just to name a few. In this Letter we experimentallysimulate the ground state of an XXZ spin chain [13] ina liquid-state nuclear magnetic resonance (NMR) quan-tum information processor [14] and directly measure a global multipartite entanglement —the geometric entan-glement (GE) [15, 16] in a version of NMR analog, orpseudo-entanglement. Exploiting the probed behaviorof GE, we identify two QPTs, which in the thermody-namic limit correspond to the first- and ∞ -orders, respec-tively. In the ∞ -order QPT, also known as Kosterlitz-Thouless (KT) transition [17, 18], the ground-state en-ergy is not singular. Consequently the detection of thecritical point in the KT transition may pose a challengefor the correlation-based approaches [5, 19], which rely onthe singularity of ground-state energy. Surprisingly, theGE turns out to be non-analytical but of different typesof singularity at the first- and ∞ -order transitions [20].Remarkably, the qualitative features of the ground-stateentanglement in the thermodynamic limit displayed nearboth transitions persist even for a small-size system, onwhich our experiment is performed.The GE of a pure many-spin quantum state | Φ i is captured by the maximal overlap [15, 16] Λ max ≡ max Ψ |h Ψ | Φ i| , and is defined as E log = − log Λ ,where | Ψ i ≡ N Ni =1 | ψ ( i ) i denotes all product (i.e. unen-tangled) states of the N -spin system. From the point ofview of local measurements, the GE is essentially (mod-ulo a logarithmic function) the maximal probability thatcan be achieved by a local projective measurement onevery site, and the closest product state signifies the op-timal measurement setting. The GE has been employedto study QPTs [16, 21], local state discrimination [22]and entanglement as computational resources [23].The XXZ spin chain is described by the Hamiltonian H XXZ = N X i =1 ( X i X i +1 + Y i Y i +1 + γZ i Z i +1 ) (1)where X i , Y i , Z i denote the Pauli matrices with i indi-cating the spin location, and γ is the control parameterfor QPTs. We use the periodic boundary condition with N + 1 ≡
1. The XXZ chain can be exactly solved by theso-called Bethe Ansatz and exhibits rich phase diagramsin the ground state [13]. In the thermodynamic limit, −2 −1 0 1 2 3−15−10−5 E g a −2 −1 0 1 2 3012 b γ Λ o r E l og FIG. 1: (Color online). Theoretical results in the 4-spin XXZchain. (a) Energy level of the ground state. (b) The overlapsquare Λ i ( γ ) and entanglement E log ( γ ). The solid, dashedand dash-dotted curves show Λ i ( γ ) for | Ψ i = | i , | Ψ i = | + − + −i and | Ψ i = | i , respectively. E log ( γ ) is shownas the dotted curve. The jump at γ = − γ = 1 in E log ( γ ) indicate the transition points for the QPTs,with the first- and ∞ - orders, respectively. for γ < −
1, the system has the ground state with theferromagnetic (FM) Ising phase. At γ = − − < γ ≤
1, the system is in a gaplessphase or XY-like phase. At γ = 1 there is an ∞ -orderor a KT transition [17], where the ground-state energy,however, is analytic across the transition, and so is anycorrelation function. For γ >
1, the system is in the N´eel-like antiferromagnetic (AFM) phase. The ground stateis asymptotically doubly degenerate. However, the exci-tations above the ground space have a gap. For γ ≫ | ... ... i , where | i ≡ | ↑i and | i ≡ | ↓i .In the XXZ chain (1), the GE displays a jump across γ = − γ = 1 [20]. Both features are present for small-size systems, as well as in the thermodynamic limit. For γ < − − < γ ≤
1, the closestproduct state is found to be | + − + − ... i , whereas for γ ≥
1, the closest product state is found to be | ... i ,where |±i ≡ ( | i ± | i ) / √ γ = 1, because of rotational symmetry, the closest prod-uct states are | φ φ ⊥ φ φ ⊥ ... i , where | φ i and | φ ⊥ i are anyarbitrary orthonormal qubit states. The singular behav-ior of ground-state GE can be used to probe the KTtransition, and there is no need to know the low-lyingspectrum.In implementation we use a 4-spin chain. The entan- glement features pertinent to the QPTs in the thermo-dynamic limit will survive. The ground-state energy andwave function of the 4-spin chain are represented as (seeSupplemental Material) E g = ( γ ( γ < − − γ − p γ + 8 ( γ > − , (2) | g i = ( | i ( γ < − | φ i cos α + | φ i sin α ( γ > − , (3)where α ∈ ( − π/ ,
0) is given via tan(2 α ) = − √ /γ , and | φ i ≡ ( | i + | i ) / √ | φ i ≡ ( | i + | i + | i + | i ) /
2. Fig. 1 (a) shows E g as a functionof γ . One should note that E g is continuous at γ = − | g i is discontinuous.In order to obtain the ground-state GE, one needs tosearch for the closest product state | Ψ ∗ i [20]. In fact,we can choose the product states | Ψ ∗ ( γ ) i = | Ψ i ≡ | i ( γ < − | Ψ i ≡ | + − + −i ( − < γ < | Ψ i ≡ | i ( γ >
1) (4)to obtain the corresponding entanglement in the respec-tive range of γ . To anticipate the experimental proce-dure, we shall measure the ground-state overlap listed asΛ ( γ ) = h Ψ | g i = ( γ < − γ > −
1) (5)Λ ( γ ) = h Ψ | g i = ( ( γ < − √ cos α − sin α ( γ > −
1) (6)Λ ( γ ) = h Ψ | g i = ( γ < − √ cos α ( γ > − . (7)From Eqs. (6) and (7), one finds that Λ ( γ ) and Λ ( γ )cross at γ = 1. Fig. 1 (b) shows the theoretical predictionfor Λ i ( γ ) ( i = 1, 2, 3) and the entanglement E log . Thejump in the entanglement at γ = − γ = 1 signify the two QPT points.In experiment, we choose the four carbons in crotonicacid [25] dissolved in d6-acetone as the four qubits. Wegenerate the ground states using quantum networks andimplement the quantum gates by GRAPE pulses [26].Various ground-states can be created by varying the sin-gle spin rotations that can be easily implemented (seeSupplemental Material). In principle, one can employan iterative method to experimentally measure the GEof the ground state (see Supplemental Material). Todemonstrate the proof-of-principle simulation of quan-tum entanglement, instead, we first measure the overlap −2 −1 0 1 2 300.51 γ Λ aa −2 −1 0 1 2 30123 γ E l og bb FIG. 2: (Color online). Experimentally measured Λ i ( γ ) (a)and E log (b) for various γ . In figure (a), the experimen-tal data are shown as ” · ”, ” × ” and ”*” for Λ i = |h Ψ i | g i| ,corresponding to | Ψ i , | Ψ i and | Ψ i . The measured Λ i ( γ )for γ < − γ > −
1, Λ ( γ ) can be fitted as thedotted line. Through fitting the points for Λ ( γ ) and Λ ( γ )using polynomial functions, we obtain the two solid curvesthat cross at point γ = 0 .
92, close to the theoretical point at γ = 1. The thick dashed and dash-dotted curves show thefitting results using the theoretical Λ ( γ ) and Λ ( γ ) by in-troducing decay factors 0 .
69 and 0 .
71, respectively. In figure(b), in range γ < − E log is shown as ”+”. For γ > − ( γ ) and Λ ( γ ) as Λ ( γ ) / .
69 andΛ ( γ ) / .
71, respectively. The dotted and dashed curves thatcross at γ = 1 .
02 show the fitting results of the rescaled datausing polynomial functions. The expected E log after rescal-ing is indicated by ” ◦ ”. of the ground state with several product states (4), whichcontain the closest product states. From the measure-ment with the already known closest product states, wecan obtain the ground-state GE. Next, to show that theobtained results are the optimum, we vary the productstates to test the optimality.The experimentally measured Λ i ( γ ) for various γ areshown in Fig. 2 (a). The measured Λ i ( γ ) for γ < − .
92, 0 .
048 and 0 . /
16 and 0, respectively. In the region γ > − ( γ ) can be fitted as Λ ( γ ) = 0 . ( γ ) andΛ ( γ ), and obtain the two solid curves that cross at the β ( π ) Λ a0 0.2 0.4 0.6 0.8 10.10.20.30.4 β ( π ) Λ b FIG. 3: (Color online). Theoretical (a) and experimen-tally measured (b) Λ for various product states created as U p ( β ) | i [see Eq. (8)] . Three ground states for γ = − . are shown asthe the solid, dash-dotted and dashed curves in figure (a),respectively. The experimental data are shown as ”*”, ” · ”and ” × ” for γ = − .
9, 1 and 3 in figure (b), respectively. Incomparison with the theoretical values, they can be fitted as0 . , 0 . and 0 . , shown as the solid, dash-dottedand dashed curves. point γ = 0 .
92, which is very close to the theoreticallypredicted transition point at γ = 1. The discrepancy be-tween experiment and theory mainly comes from the dif-ferent experimental errors in measuring Λ ( γ ) and Λ ( γ ).The jump at γ = − γ = 0 .
92 reflect thedifferent types of QPT points.In order to faithfully estimate the performance ofthe experiment in measuring Λ ( γ ) and Λ ( γ ) in therange γ > −
1, we introduce two decay factors α and α to fit the experimental data as [Λ , ( γ )] exp = α , [Λ , ( γ )] theory , shown as the thick dashed and dash-dotted curves in Fig. 2 (a) with the best scale-factorsas α = 0 .
69 and α = 0 .
71, respectively. The differ-ence between the decay factors comes from the differentoperations in measuring Λ ( γ ) and Λ ( γ ). In Fig. 2(b), we exploit the decay factors to rescale experimen-tal values of [Λ , ] exp /α , , from which we obtain the ex-pected values of pseudo-entanglement shown as ” ◦ ”. Therescaled − log ([Λ , ] exp /α , ) can be fitted as the dottedand dashed curves that cross at γ = 1 . | Ψ( β ) i by | Ψ( β ) i = U p ( β ) | i , (8)where U p ( β ) = N j =1 e − iβY j / , and experimentally mea-sure Λ = |h Ψ( β ) | g i| for various β at three different lo-cations of the phase diagram, corresponding to γ = − . . , 0 . and 0 . , shown in Fig. 3 (b). Onefinds that the maximum of Λ occurs at β = π/ γ = − . | Ψ i = | + − + −i and | Ψ i = | i , predicted theoretically. Remarkably,for γ = 1, where the ∞ -order QPT occurs, Λ is a con-stant independent of β , as we have expected and notedearlier. This also means that arbitrary states preparedby Eq. (8) can be chosen to measure the entanglementat γ = 1, and this gives additional confirmation that thecreated state at the KT point is rotationally invariant.The experiment duration of the preparation of theground states for γ > − T . Consequently the decay of the signals due to the limitation of coherence time is one of main sources oferrors. Additionally the imperfection of pulses and inho-mogeneities of magnetic fields also contribute to errors.The deviations of the experimental data from the the-oretical fitting in Fig. 3 (b) represent the effects of theerrors that depend on the rotation angles, or the productstates. In particular the fluctuation of the data for γ = 1in Fig. 3 (b) confirms the explanation for the shift of themeasured cusp in Fig. 2 (a).In conclusion we demonstrate the non-analytic proper-ties of many-body systems in a quantum simulator usingNMR. The QPTs with first- and ∞ -orders in the XXZspin chain are detected by directly measuring the pseudo-entanglement of the ground states created by quantumgates. An alternative approach for creating ground stateswould be via adiabatic evolution [27]. Our prelimi-nary numerical analysis indicates that ground states for γ > − > . γ . The experimental implemen-tation is a possible future direction.We thank O. Moussa and R. Or´us for helpful dis-cussions. This work was supported by CIFAR (R.L.),NSERC (J.-F.Z., R.L. and T.-C.W.), MITACS (T.-C.W.), SHARCNET (R.L.), and QuantumWorks (R.L.). [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] M. Nielsen and I. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[3] R. Horodecki et al., Rev. Mod. Phys. , 865 (2009).[4] S. Sachdev, Quantum Phase Transitions (Cambridge Uni-versity Press, Cambridge, England, 2000).[5] A. Osterloh et al., Nature (London) , 608 (2002).[6] S. Sachdev, Nature Phys. , 173 (2008).[7] L. Amico et al., Rev. Mod. Phys. , 517 (2008).[8] O. G¨uhne and G. T´oth, Physics Reports , 1 (2009).[9] P. Horodecki, Phys. Rev. Lett. , 167901 (2003); H. A.Carteret, ibid. , 040502 (2005).[10] S. P. Walborn et al., Nature , 1022 (2006); N. Kieselet al., Phys. Rev. Lett. , 260505 (2008).[11] O. G¨uhne, M. Reimpell and R.F. Werner, Phys. Rev.Lett. , 110502 (2007).[12] R. P. Feynman, Int. J. Theor. Phys. , 467 (1982); S.Lloyd, Science , 1073 (1996); I. Buluta and F. Nori, ibid. , 108 (2009).[13] V. E. Korepin et al., Quantum Inverse Scattering Methodand Correlation Functions (Cambridge University Press,Cambridge 1997).[14] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. , 1037 (2004); J. A. Jones, Prog. in NMR Spectrosc. ,325 (2001).[15] T.-C. Wei and P. M. Goldbart, Phys. Rev. A , 042307(2003).[16] T.-C. Wei et al., Phys. Rev. A , 060305(R) (2005).[17] J. M. Kosterlitz and D. J. Thouless, J. of Phys. C , 1181-1203 (1973).[18] C. C. Rulli and M. S. Sarandy, Phys. Rev. A , 032334(2010).[19] L.-A. Wu et al., Phys. Rev. Lett. , 250404 (2004); S.-J.Gu, Int. J. Mod. Phys. B , 4371(2010).[20] R. Or´us and T.-C. Wei, Phys. Rev. B. , 155120 (2010).[21] R. Or´us, Phys. Rev. Lett. , 130502 (2008); R. Or´uset al., ibid. , 025701 (2008).[22] M. Hayashi et al., Phys. Rev. Lett. , 040501 (2006).[23] D. Gross et al., Phys. Rev. Lett. , 190501 (2009).[24] Here | + − + − ... i is equivalent to | − + − + ... i or moregenerally | θ + θ − θ + θ − ... i with θ ± ≡ ( | i ± e iθ | i ) / √ | ... i is equivalent to | ... i .[25] E. Knill et al., Nature (London) , 368 (2000).[26] N. Khaneja et al., J. Magn. Reson. , 296 (2005); C.A. Ryan et al., Phys. Rev. A , 012328 (2008).[27] P. Kr´ a l, I. Thanopulos and M. Shapiro, Rev. Mod. Phys. , 53 (2007); X. Peng et al., Phys. Rev. Lett. , 140501(2009). Supplemental Material
Appendix A: Method for computing and measuringthe maximal overlap by iteration
Here we describe an iterative method to computethe maximal overlap, of which the motivation comesfrom the density-matrix-renormalization-group (DMRG)or matrix-product-state (MPS) variational method [1].This method can not only be implemented numerically,but can also be carried out experimentally. To computethe maximal overlap for the state | g i with respect toproduct states | Ψ i ≡ N Ni =1 | ψ ( i ) i , we use the Lagrangemultiplier λ to enforce the constraint h Ψ | Ψ i = 1, f (Ψ) ≡ h Ψ | g ih g | Ψ i − λ h Ψ | Ψ i . (A1)Maximizing f with respect to the local product state | ψ ( i ) i , we obtain the extremal condition H ( i )eff | ψ ( i ) i = λN ( i ) | ψ ( i ) i , (A2)where H ( i )eff ≡ ( N Nj = i h ψ ( j ) | ) | g ih g | ( N Nj = i | ψ ( j ) i ) is propor-tional to a local projector (labeled by | φ ( i ) ih φ ( i ) | at i -site), and the normalization N ( i ) ≡ N Nj = i h ψ ( j ) | ψ ( j ) i , isunity if all the local states are properly normalized. Fromthe viewpoint of the variational MPS, one fixes all localstates | ψ ( j ) i but | ψ ( i ) i and solves for the correspondingoptimal | ψ ( i ) i and repeats the same procedure for i + 1, i + 2, etc. until the N -th site and sweeps the procedureback and forth until the eigenvalue λ converges. Theconverged value | λ | is the square of the maximal over-lap Λ .Experimentally, this procedure means that one choosesrandomly the local measurement basis and picks arbitrar-ily one direction (i.e., rank-one projector) for each site,say, | ψ ( j ) i for the j -th site and only varies the basis forone site, say, i -th at a time with all others fixed until onereaches a basis where the measurement outcome alongone direction occurs with the most probability. Then onemoves to the next site, say, ( i + 1)-th site, and finds theoptimal direction and repeats this procedure by sweepingback and forth until the probability for the most likelyoutcome converges.Numerically, it appears that one has to solve the aboveeigenvalue problem. But as the effective local Hamilto-nian is a projector, one immediately sees that the optimalrank-one projector is exactly the one ( | φ ( i ) ih φ ( i ) | ) givenby H ( i )eff . One thus replaces | ψ ( i ) i by | φ ( i ) i and repeats theprocedure at other sites until the overlap converges. Ex-perimentally, the apparatus setting in line with the pro-jector gives the local maximum output probability. Thesearch for the optimal direction at the i -th site need notbe a blind search, as a tomography (conditioned on allother sites being measured in their respective | ψ ( j ) i ) willenable the determination of the optimal local direction | φ ( i ) i . The whole procedure, either numerically or ex-perimentally, thus becomes an iterative procedure, givenan initial choice of {| ψ ( j ) i} . We have performed such anumerical procedure and have found that this procedureconverges efficiently to the maximal overlap. The conver-gence for the ground state of four-spin chain is very fastand the result is accurate; see Fig. 4. In our experiment,we implement a simplified version to directly measure theground-state entanglement. Λ FIG. 4: (Color online). Numerical implementation of theiteration procedure for the maximal overlap in the 4-spin XXZchain. The overlap square is shown as marked as ”*”, ” · ” and” × ” for γ = − .
9, 1 and 3, respectively. An initial 4-spinproduct state is randomly chosen. One round of sweep startsfrom the first spin, reaches the forth spin, and sweeps backto the second spin. One round thus contains six steps. Eightiteration rounds are shown. The maximal overlap squaresconverge rapidly to their exact values, which are indicated bythe thin lines, after just a few rounds. We have checked thatthe converged product states are | + − + −i (up to a relativephase), | φφ ⊥ φφ ⊥ i (with h φ ⊥ | φ i = 0), and | i , respectively. Appendix B: Method for solving the 4-spin XXZchain
For γ < − | i and | i . To avoidthe complication due to the degeneracy, we can intro-duce an additional small Zeeman term H z = B z P Ni =1 Z i with 0 < B z ≪ | i so that the ground state | g i becomes | i .The small Zeeman term does not affect the universalityclass of phase transition of the ground state, because itcommutes with H XXZ .For γ > −
1, the ground state is not a simple productstate. Due to the absence of an external field, the conser-vation of z-component total angular momentum and theperiodic translation invariance give rise to only two rele-vant “Bethe-ansatz” basis states for the ground states: | φ i ≡ √ | i + | i ) (B1) | φ i ≡
12 ( | i + | i + | i + | i ) . (B2)By solving the effective Hamiltonian in the subspacespanned by {| φ i , | φ i} H eff = − (cid:18) γ −√ −√ (cid:19) (B3)we obtain that the ground state is | g i = | φ i cos α + | φ i sin α (B4)where α ∈ ( − π/ ,
0) is given viatan(2 α ) = − √ /γ, (B5)and that the ground energy is E g ( γ ) = − γ − p γ + 8 . (B6) Appendix C: Method for experimentalimplementation
The experiment is performed in a Bruker DRX 700MHz spectrometer. The structure of the molecule of cro-tonic acid and the parameters of the four spin qubits areshown in Fig. 5 (a). The protons are decoupled in thewhole experiment. The initial pseudo-pure state | i is prepared by spatial averaging [2, 3], and chosen as thereference state for normalizing the signals in the followingpartial state tomography.The quantum circuit shown as Fig. 5 (b) illustratesthe experiment protocol for γ > −
1. The ground stateis created by U U ( α ) U , indicated by the three dashedblocks, respectively. We optimize U and U , which areindependent of α , as two long (40 ms duration) GRAPEpulses, respectively, where GRAPE stands for gradientascent pulse engineering. The theoretical fidelity for U and U is larger than 99%. To save time in searchingGRAPE pulses, we further decompose U ( α ) into simplegates shown as the sequence in Fig. 5 (c), where eachgate is implemented by one GRAPE pulse. The durationof the pulse for the spin coupling evolution is 20 ms, andthe duration of other pulses is 0.5 ms. The theoreticalfidelity for each pulse in Fig. 5 (c) is larger than 99 . γ > − α in the single spin operation, which ismuch easier to find than U ( α ) in the GRAPE algorithm.For the case of γ < −
1, we replace U U ( α ) U by fourNOT gates implemented by four π pulses applied to thefour qubits respectively to create the ground state | i from | i .To measure the overlap between | g i and an arbitraryproduct state | Ψ i , we re-write the overlap Λ ≡ h Ψ | g i inform of Λ = h b | U † p | g i (C1)where | b i denotes a computational basis and U p | b i = | Ψ i [3]. Here we choose U p as U p ( β ) = O j =1 e − iβY j / . (C2)Since | Ψ i and | Ψ i are already the computational basis,we can simply choose | b i as | Ψ i and | Ψ i , respectively,and take U p as the identity operation by setting β = 0,for obtaining Λ and Λ from Eq. (C1). | Ψ i is not acomputational basis. We can choose | b i = | i and β = π/ , noting that | Ψ i = U p ( π/ | i .In the density-matrix form, Eq. (C1) is represented asΛ = T r ( | b ih b | ρ ) (C3)where ρ = U † p ( | g ih g | ) U p . From Eq. (C3), one finds thatΛ is encoded as the diagonal element | b ih b | of ρ . Weexploit phase cycling to remove all the non-diagonal ele-ments, and then reconstruct all the diagonal terms of ρ using partial state tomographgy through four π/ is there-fore extracted from the diagonal terms. [1] F. Verstraete et al., Adv. Phys. , 143 (2008).[2] J. S. Hodges et al., Phys. Rev. A , 042320 (2007). [3] J. Zhang et al., Phys. Rev. A , 012305 (2009). T (s) O cb =exp(-i Z Z /4) qyyyy (-1) n (-1) m (-1) k zzz U yyxyxxy U ( ) |0> |0> |0> H * H R - XXH R |0> U x U +p D (-1) j z t o m og r aph y =exp(i Q/2) a G r ound s t a t e p r epa r a t i on R o t a t i on D epha s i ng M ea s u r e m en t =CNOT by |1> =CNOT by |0>=controlled O by |1>O=controlled O by |0> Legend (q=x,y,z;Q=X,Y,Z)
FIG. 5: (Color online). Experimental protocol. (a) Themolecule structure (inset) and the parameters of the fourcarbon-13 spins. The diagonal terms show the chemical shiftsand the non-diagonal terms show the strength of J - couplingsin Hz. The transversal relaxation times T measured by aHahn echo are listed in the rightmost column. (b) Quan-tum circuit for creating the ground state and measuring itsoverlap Λ i ( γ ) for γ > −
1. Here X denotes a NOT gate. R ± α = exp ( ± iαY ), and H ( H ∗ ) denotes a gate transform-ing state | i ( | i ) to ( | i + | i ) / √
2, and | i ( | i ) to ( | i - | i ) / √
2, respectively. The other gates are illustrated in thelegend box. U is decomposed as the gate sequence shown in(c), for creating an arbitrary ground state for γ > − α [see Eq. (B5)]. U p denotes a rotation for obtainingthe overlap of the ground-state with U p | b i , where | b i denotesa computational basis. In dephasing operation, j , k , m , n are chosen as 0 and 1, respectively, to average out all non-diagonal terms in the density matrix to zero. Tomograhpy ofthe diagonal terms requires four π/π/