Measuring Holographic Entanglement Entropy on a Quantum Simulator Keren Li,1, 2, ∗ Muxin Han,3, 4, ∗ Guilu Long,1 Yidun Wan,5, 6, † Dawei Lu,2, 7, ‡ Bei Zeng,2, 7, 8, 9, § and Raymond Laflamme2, 9, 10
arXiv:1705.00365v1 [quant-ph] 30 Apr 2017
1
State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China 2 Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada 3 Department of Physics, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA 4 Institut f¨ur Quantengravitation, Universit¨at Erlangen-N¨urnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany 5 Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China 6 Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China 7 Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 8 Department of Mathematics and Statistics, University of Guelph, Guelph N1G 2W1, Ontario, Canada 9 Canadian Institute for Advanced Research, Toronto M5G 1Z8, Ontario, Canada 10 Perimeter Institute for Theoretical Physics, Waterloo N2L 2Y5, Ontario, Canada Anti-de Sitter/conformal field theory (AdS / CFT) correspondence is one of the most promising realizations of holographic principle towards quantum gravity. The recent development of a discrete version of AdS/CFT correspondence in terms of tensor networks motivates one to simulate and demonstrate AdS/CFT correspondence on quantum simulators. We achieve this goal indeed, in this work, on a six-qubit nuclear magnetic resonance quantum simulator. We demonstrate experimentally the discrete AdS/CFT correspondence, under realistic noises, by measuring the relevant entanglement entropies on the corresponding tensor network state. The fidelity of our experimentally prepared tensor network state is 85.0% via full state tomography and rises to 93.7% if the signaldecay due to decoherence is taken into account. Our experiment serves as the basic module of simulating more complex tensor network states that exhibit AdS/CFT correspondence. As the initial experimental attempt to study AdS/CFT via quantum information processing, our work opens up new avenues exploring quantum gravity phenomena on quantum simulators. PACS numbers:
Anti-de Sitter/conformal field theory (AdS / CFT) correspondence is one of the most prominent approaches towards a quantum theory of gravity for over two decades [1, 2]. It is the most successful realization of the holographic principle to date, by stating that the quantum gravity theory in the bulk anti-de Sitter spacetime is actually equivalent to a conformal field theory on the lower-dimensional boundary of the spacetime. Beyond its applications in quantum gravity and unified theory of all fundamental interactions, AdS/CFT is also widely used in strongly coupled quantum systems, quantum chromodynamics, and condense matter physics [3–5]. Furthermore, it has recently become a bridge connecting quantum gravity to quantum information theory [6, 7], which inspires revolutionary ideas of developing quantum gravity using the methods in quantum information and entanglement. Within this new framework, quantum gravity can be understood via exactly solvable quantum models. For instance, a black hole is related to quantum scrambling in the chaotic systems [8–10] and studied in the Sachdev-Ye-Kitaev model [11–13]; the spacetime geometry emerges from entanglement [14, 15], which can be realized by tensor networks and quantum error correcting codes [16–19]. The holographic duality of AdS/CFT offers a dictionary between the observables of the d-dimensional bulk gravity theory and those of the (d−1)-dimensional boundary field theory, i.e. properties of the bulk gravity and geometry can be reconstructed or emerged from the boundary field theory (known as the emergent gravity program [15]), and vice versa. In particular, the bulk geometry can be reconstructed holographically
from the entanglement in the boundary field theory, characterized by the Ryu-Takayanagi (RT) formula S EE (A) =
Armin , 4G N
(1)
which identifies the entanglement entropy S EE (A) of a (d−1)dimensional boundary region A with the area Armin of the bulk (d − 2)-dimensional minimal surface anchored to A [14, 20, 21]. G N is the Newton constant. See Fig. 1(a) for a brief illustration. Recently, a discrete version of AdS/CFT is realized on a type of lattices called tensor networks (TN) [17, 18, 22, 23], making it possible to be demonstrated on a quantum simulator device in practice. In general, TN states are ways of rewriting a manybody wave function in terms of contractions of tensors, aiming at obtaining the ground states of interacting many-body Hamiltonians in a numerically efficient way. As a key observation related to AdS/CFT , a TN state has an emergent bulk dimension built by the layers of tensors, making it an ideal ground for manifesting AdS/CFT in many-body systems. In other words, gravity/geometry is anticipated to be created in the emergent bulk dimension of TN. Indeed, this anticipation is fulfilled by the TN made of perfect tensors, which exhibit interesting holographic properties. In particular, perfect tensors (PT) give exactly the holographic entanglement entropy realizing the RT formula [17]. In this work, we demonstrate the RT formula and hence A d S / CFT on a quantum simulator that simulates the minimal PT of rank-6. Using a six-qubit quantum register in the nuclear magnetic resonance (NMR) system, we create the rank-
2 6 PT and subsequently measure its holographic entanglement entropy. The experimental results demonstrate the RT formula if the decoherence effect is taken into account. As the rank-6 PT serves as the building block to construct the entire TN , our experiment also opens up a new and practical way of studying A d S / CFT and the holographic principle at large. Perfect tensors – The TN that we focus on is shown in Fig. 1(b), where each hexagon represents a special six-qubit state |ψi. |ψi is called a PT, if and only if that any three-qubit subsystem out of six is maximally entangled with the rest. It is shown that, for a TN made by the PT, its entanglement entropy is holographic and gives the discrete RT formula on the lattice. Actually, the entanglement entropy of such TN equals the minimal number of links cut by the virtual surfaces anchored to the boundary, as illustrated in Fig. 1(b). (a)
(b)
-
-
n
Bulk qubits
(c)
2
2
3 1
1
3
6
4
4 5 6
5 Boundary qubits
FIG. 1: (a) A sketch of the RT formula. The hexagonal tiling indicates that the disk is a 2-dimensional AdS space. The red solid arc in the bulk is the minimal surface (a line in this case) anchored to the two ends of a chosen boundary region A. (b) A discretization of (a) by a tensor network comprised of rank-6 tensors. Each hexagonal node represents a rank-6 tensor state |ψi ∈ H ⊗6 , and the collection of all such nodes corresponds to the tensor product of all |ψi’s. Each √ link ` represents a maximally entangled state |`i = (|00i + |11i) / 2. Connecting one leg of the node to a link corresponds to taking the inner product in H. The dangling legs are physical qubits in the many-body system. The red dashed arc illustrates the virtual surface S anchored to region A, which cuts a minimal number of links. (c) Rank-6 PT from the TN with the minimal number of cuts equal to three. The six legs represent six qubits. Three qubits are at the boundary and the other three are bulk qubits. This is the model realized in our experiment.
To prove the above statement, we first introduce the form of the rank-6 PT, which is the building block of the TN. Given the single-qubit Hilbert space H ' C2 , a rank-6 PT |ψi is a state in H ⊗6 , such that for any bipartition of qubits m + k = 6, the
entropy of the reduced density matrix is maximal. Assuming m ≥ k, and labeling the orthonormal basis in H ⊗m and H ⊗k P by |αi and |ii respectively, a PT |ψi = α,i ψαi |αi ⊗ |ii satisfies X
ψ† iα ψα j =
α
1 δi j . 2k
(2)
In other words, the reduced density matrix ρ(k) by tracing out m qubits is an identity matrix, whose entanglement entropy (k) (k) S (k) EE = −tr ρ log2 ρ
(3)
is simply k, the number of remained qubits. In this Letter, we use the superscript (k) to represent the k-qubit subsystem. With the rank-6 PT (explicit form in appendix A [24]) in hand, the TN state illustrated in Fig. 1(b) is constructed as follows. Each internal link ` represents √ a two-qubit maximally entangled state |`i = (|00i + |11i) / 2, where two qubits associate respectively to the two end points of `. If we denote by |ψ(n)i the PT associated to the hexagon node n, the total TN state |Ψi in Fig. 1(b) is written as a (partial) inner product form O O |Ψi = h`| |ψ(n)i. (4) `
n
The inner product takes place at the end points of each internal link `, between one qubit in |`i and the other in |ψ(n)i. The qubits in |ψ(n)i not participating the inner product are boundary qubits corresponding to the dangling legs, and these boundary ones are actually physical qubits, indicating that |Ψi is a state on the boundary. We then pick a boundary region A which collects a subset of the boundary qubits, as shown in Fig. 1(b). The reduced density matrix ρA = trA¯ (|ΨihΨ|) is computed by tracing out all boundary qubits outside A. Initially, this partial trace boils down to computing the reduced density matrix of individual tensors closest to the boundary. By applying Eq. (2) and noticing that |`i is maximally entangled, the trace computation can be effectively pushed from the boundary into the bulk, meaning that the partial trace on the boundary is now equivalent to computing the reduced density matrix of the PT inside the bulk [24]. Once again, we can apply Eq. (2) and push the trace further inside. This iteration procedure is repeated until the trace reaches S in Fig. 1(b), where Eq. (2) is not anymore valid, as the number of qubits participating the trace (number of links cut by S) is less than three for each tensor. Now we have presented a sketch about how to calculate the entanglement entropy of ρA via Eq. (3), and direct readers to appendix B [24] for a concise proof using the graphical computation of TN. Firstly, tr (ρA ) is found to be equal to the number of qubits on S, i.e. the same as the number of links cut by S. Moreover, the product ρ2A , involving the inner product of boundary qubits in A, gives that ρ2A ∝ ρA . Note that we have ignored all numerical prefactors but they all cancel when trρn calculating (trρAA)n in the entanglement entropy. As a result, the
3 (a)
Von Neumann entropy gives [24] trρnA
1 log2 n→1 1 − n (trρA )n = minimal number of cuts by S.
S EE (A) = lim
(5)
The above result is a discrete version of the RT formula in Eq. (1). The “minimal number of cuts” in Eq. (5) represents the minimal area Armin (in the unit of Planck scale) in the RT formula. The bulk surface S with minimal area emerges effectively from the entanglement entropy of the TN state. Eq. (5) demonstrates explicitly that the bulk geometry are created holographically by the entangled qubits of the boundary many-body system. It is worth emphasizing that, all descriptions about constructing the TN originate from the PT in Eq. (2). Therefore, this rank-6 PT plays the fundamental role in holographic entanglement entropy, and is a key of emerging bulk gravity from TN states. If we choose S as shown in Fig. 1(c) by which the minimal number of cuts is three, a rank-6 PT is generated where the boundary and bulk qubits are both three. In this work, we demonstrate the emergent gravity program in A d S / CFT for the first time in a six-qubit NMR quantum simulator, by creating the rank-6 PT in Fig. 1(c) and measuring the relevant entanglement entropies. Now we turn to the experimental part. Experiment implementation of a rank-6 perfect tensor – The six qubits in the NMR quantum register are denoted by the spin-1/2 13 C nuclear spins, labeled as 1 to 6 as shown in Fig. 2(a), in 13 C-labeled Dichloro-cyclobutanone dissolved in d6 -acetone. All experiments were carried out on a Bruker DRX 700 MH z spectrometer at room temperature. The internal Hamiltonian of this system is Hint =
6 X j=1
πν j σzj +
6 X π J jk σzj σkz , 2 j 3, there are notable discrepancies between theory and experiment, which should be primarily attributed to decoherence errors, as discussed in the following. The pulse sequence that creates the PT is around 60 ms [24]; this is not a negligible length compared to the T 2∗ time (∼ 400 ms) of the molecule, meaning that decoherence will induce substantial errors during experiments. As T 2∗ relaxation is the dominating factor, the off-diagonal terms in the PT density matrix are mainly affected. To estimate this imperfection, we performed full state tomography [25] on the prepared state and got ρe . The real part of ρe is depicted in the right panel of Fig. 4(a), by projecting each element onto a two-dimensional plane. As a comparison, the figure of the theoretical PT ρ pt = |ψihψ| is placed in the left panel of Fig. 4(a). In fact, the diagonal elements of ρe are almost the same as that of ρ pt , but the off-diagonal are lower due to the T 2∗ errors. The state fidelity between ρe and ρ pt , defined as q √ √ F ρ pt , ρe = tr ρ pt ρe ρ pt , (7) is about 85.0%. Direct observations of ρe in terms of NMR
4 5
S iden�ty S PT S exp S exp_nodec
3
�ty Iden
S
(k)
4
2
PT
1 1
2
3
k
4
5
FIG. 3: Entanglement entropy S (k) of the k-qubit subsystem of the rank-6 PT. In theory, S (k) = min{k, 6 − k} as shown by the orange dashed line. Experimental results are represented by the red circles, where S (4) and S (5) do not fit very well. If the signal’s decay due to decoherence is taken into account, the experimental results are rescaled to the blue squares, which fit much better. As a upper-bound reference, the maximal entropy of a k-qubit subsystem is also plotted (green dotted line) by assuming a six-qubit identity.
PT —a build block of a complex TN, we have achieved 85% fidelity, which is already state-of-the-art; however, there is yet some non-negligible decoherence due to the T 2∗ errors. Therefore, our results successfully test the RT formula up to the decoherence. Our results further suggest that a larger quantum processor would be able to simulate higher dimensional RT formulae and hence the AdS/CFT correspondence, under either of the two assumptions: the decoherence effect during experiments can be tolerated, or the original TN state is supposed to be pure. The latter has been in fact tested and elaborated in detail in [28].
(a)
1 2/3 1/3 0 -1/3 -2/3 -1
10
(b) NMR signal (a.u.)
spectra are also shown in Fig. 4(b), where experimental and simulated spectra highly match if the experimental signal is rescaled by 1.25 times to compensate for the decoherence effect. Although the reconstructed state ρe is prone to the decoherence errors, the entanglement entropies for the cases k ≤ 3 in Fig. 3 are still in excellent accordance with the theory. The reason is, when we trace out three or more qubits, the reduced density matrix is predicted to be identity according to Eq. (2), so the measured k ≤ 3 reduced density matrices are almost irrelevant to the imperfection of the off-diagonal elements in ρe . However, when k > 3, the reduced density matrix is no longer the identity, meaning that the imperfect off-diagonal terms in ρe start to be responsible for calculating S (k) . As a result, in Fig. 3 we have S (4) = 2.91 ± 0.20 and S (5) = 2.32 ± 0.25 (red circles) respectively, which are quite distant from the theoretical curve. After numerically simulating and compensating for the decoherence errors [26, 27] during the PT creation, we found that the two entanglement entropies S (4) and S (5) approach much closer to the theory, which are now 2.27 ± 0.46 and 1.37 ± 0.28 (blue squares), respectively. We also calculated the current fidelity between the rescaled experimental state and ρ pt via Eq. (7), and found it improved to 93.7%, which is 8.7% greater than that of ρe . Discussion – RT formula, or explicitly the TN built by the rank-6 PT in Fig. 1(b), tells us how to deduce the bulk geometry using the entanglement on the boundary. The implicit condition here is that the global TN state is pure. Otherwise, the information on the boundary cannot uniquely (up to local unitaries) determine the bulk geometry, e.g. it cannot specify whether the TN state is the maximally mixed identity or PT since both give the same entanglement entropies on the boundary (meaning k ≤ 3) as shown in Fig. 3. In experiments, however, under realistic noises, it is difficult to guarantee the purity of the truly created states because experimental procedures inevitably involve errors—in particular the decoherence that render the TN states mixed. In our experiment of a 6-qubit
1
C1
exp
sim
0 -1
-80 1
-60
-40
-20
0
20
40
C4
60 exp
80 sim
0 -1
-80
-60
-40
-20
0
20
40
60
80
Frequency (Hz)
FIG. 4: (a) Density matrices of the theoretical rank-6 PT ρ pt (left) and the experimentally reconstructed state ρe (right) on a twodimensional plane. The rows and columns are labeled by the sixqubit computational basis from |0i⊗6 to |1i⊗6 , respectively. (b) Direct observation of ρe in the NMR spectra (red), with probe qubits C1 (top) and C4 (bottom), respectively. The simulated spectra of the PT are also shown in blue. For a better visualization, experimental signals are rescaled by 1.25 times to neutralize the decoherence error.
Conclusion – Our work is the first endeavor to demonstrate
RT formula in A d S / CFT correspondence using the discrete per-
fect tensor version on a quantum simulator. We utilize a temporal average technique to create the rank-6 PT, and perform full state tomography to reconstruct the experimental state. This is also the largest full state characterization in a spin ensemble system to date. Although the imperfection of the created state due to decoherence errors makes the holographic entanglement entropy not exactly agree with the theoretical prediction, we simulate and compensate for such type of errors under the realistic experimental environment, and demonstrate the accordance between theory and experiment thereafter. As the first step towards exploring AdS/CFT correspondence using a quantum simulator, our work provides valid experimental demonstrations about studying quantum gravity in the presence of realistic noises.
5 Acknowledgements – We thank Jianxin Chen, Markus Grassl, Cheng Guo, Ling-Yan Hung, Zhengfeng Ji, Hengyan Wang, and Nengkun Yu for helpful discussions. This research was supported by CIFAR, NSERC and Industry of Canada. K.L. and G.L. acknowledge National Natural Science Foundation of China under Grants No. 11175094 and No. 91221205. MH acknowledges support from the US National Science Foundation through grant PHY-1602867, and Start-up Grant at Florida Atlantic University, USA. YW thanks the startup grant offered by the Fudan University and the hospitality of IQC and PI during his visit, where this work was partially conducted.
∗ † ‡ §
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[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
These authors contributed equally to this work. Electronic address:
[email protected] Electronic address:
[email protected] Electronic address:
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Supplemental Information for “Measuring Holographic Entanglement Entropy on a Quantum Simulator” A.
Perfect tensor
A n-qudit state |ψn is perfect if for any bipartition, the entropy of the reduced state is maximal. For example, in the case of n = 2, any maximally entangled state is perfect. Moreover, in a two-level (qubit) system, the perfect tensor state only exists in the Hilbert space of two, three, five, and six qubits. Any GHZ state is perfect for n = 3; any state of the five-qubit error correcting code is perfect for n = 5 [1]; for n = 6, the perfect state is unique, up to local unitaries (LU) and permutations [2]. One representation of the six-qubit perfect state is given by the graph state with stabilizers in Table. S1. It can also be expanded in terms of computational basis or Pauli basis only with a difference of LU or permutation as shown in Table. S1. This state is just what we have prepared in experiment. B.
Derivation of the RT formula
Fig. S1 is a graphical presentation of Eq. (2) being the defining property of the perfect tensor (PT). In Fig. S1, the white hexagon node with six legs represents a rank-6 PT |ψ = ψa1 ···a6 |a1 ⊗ · · · |a6 (1)
=
=
= FIG. S1. Graphical presentation for the properties of PTs. Each figure on the LHS represents a contraction of two rank-6 PTs: an internal line between two PTs implies a common index of the two tensors that is summed over all its possible values. Each line on |aa| in the the RHS represents a maximally entangled state a corresponding 2-qubit Hilbert space, where each qubit sits at an end of the line.
region
a1 ,··· ,a6
in H , where |ai ∈ H is an orthonormal basis. The black hexagon represents the Hermitian conjugate ψ|. Connecting a pair of legs between two hexagon nodes means contracting the a pair of tensor indices between ψa1 ···a6 and its conjugate. Equivalently, it means to have inner product between |ψψ| and the maximally entangled state ai |ai ai | in the two-qubit H ⊗ H∗ space (or identity operator in H): ai ai |ψψ|a i . Each line on the right is an maximally entangled state a |aa|. Given the tensor network state |Ψ, we pick a boundary region A which collects a subset of the boundary qubits. For convenience, the region A we choose here is the same as the one in Fig. 1b in the main text. The reduced density matrix ρA = trA¯ (|ΨΨ|), tracing out all boundary qubits outside A, can be computed graphically as shown in Fig. S2, by using the rules in Fig. S1. Initially, this partial trace boils down to contracting the indices of tensors closest to the boundary. By applying Fig. S1, the tensor contractions are effectively pushed from the boundary into the bulk (see Fig. S2). Once again, we can apply Fig. S1 and push the contractions further inside. This iteration procedure is repeated until the contractions reach S (the red dashed line in Fig. S2), where Fig. S1 cannot be applied anymore, as the number of qubits participating the trace (number of links cut by S) is less than three for each tensor. Similarly, ρ2A is computed in Fig. S3, which clearly gives 2 ρA ∝ ρA by the contractions of tensors. tr(ρ2A ) and tr(ρA ) ⊗6
=
region region
region
=
region region
= ρA
FIG. S2. Graphical computation of the reduced density matrix ρA . In the first step, all boundary qubits outside A are traced out following the rules in Fig. S1. Then, the remaining two hexagon nodes in the middle are further traced out, indicating that the contractions can be pushed inside.
are computed in Fig. S4. Note that in the graphical computation, we have ignored all numerical prefactors. However, trρn the numerical prefactors we ignore all cancel in (trρAA)n when computing the entanglement entropy. As a result, the second Renyi entropy S2 (A) trρ2 − ln2 (trρAA)2 gives S2 (A) = Minimal Number of Cuts by S.
=
(2)
which equals to 3 in the example shown in the graphical computation (dim(H) = 2). The n-th Renyi entropy can be computed similarly, and gives the same result.
2
PT
000000 000001 000010
"
000011 000100
000110
000111 001000
001001
010010 010011
010100
011000 011001 011010 011011
011100
010101 010110 010111
101000 101001
100011
011101
100100 100101 100110 100111
101010
101011 101100 101101
101110 101111
110011 110100
110101 110110 110111
111000 111001 111010 111011
111100 111101 111110
001110 001111 010000 010001
110010
"
001010 001011 001100 001101
011110 011111 100000 100001 100010
PT
000101
110000
110001
111111
IIIIII+IIXYYZ-IIYXXY+IIZZZX+IXIYZY+IXXIXX-IXYZYI+IXZXIZ-IYIXYX-IYXZIY+IYYIZZ IYZYXI +IZIZXZ+IZXXZI-IZYYIX+IZZIYY+XIIZYY+XIXXIX-XIYYZI+XIZIXZ+XXIXXI XXXZZZ-XXYIIY-XXZYYX-XYIYIZ-XYXIYI-XYYZXX+XYZXZY+XZIIZX-XZXYXY +XZYXYZ+XZZZII-YIIYXX-YIXIZY+YIYZIZ-YIZXYI-YXIIYZ-YXXYII-YXYXZX+YXZZXY +YYIZZI-YYXXXZ+YYYYYY+YYZIIX-YZIXIY+YZXZYX +YZYIXI+YZZYZZ+ZIIXZZ +ZIXZXI-ZIYIYX+ZIZYIY+ZXIZIX-ZXXXYY+ZXYYXZ+ZXZIZI-ZYIIXY+ZYXYZX+ZYYXII +ZYZZYZ+ZZIYYI+ZZXIIZ+ZZYZZY-ZZZXXX
TABLE S1. Form of the rank-6 PTS in three representations: the graph state with stabilizers, computational basis, and Pauli basis. The normalization factor is ignored.
region
ρ2A =
region region
region region
=
=
dim(H3)
region
=
region region
FIG. S3. Graphical computation of Fig. S2 and the rules in Fig. S1.
C.
ρ2A
based on the result of ρA in
Preparation of the rank-6 perfect tensor state
Our six-qubit processor is per-13 C labeled (1S,4S,5S)7,7-dichloro-6-oxo-2-thiabicyclo[3.2.0]heptane-4-carboxylic acid dissolved in d6-acetone as shown in Fig. S5. It consists of seven 13 C and five 1 H spin-1/2 nuclear spins. The 1 H channel are decoupled throughout the experiments. The experimental preparation of the rank-6 PT is based on the temporal averaging method. As shown in Table. S1, the PT consists of 64 Pauli terms with the same amplitudes when
FIG. S4. Graphical computation of tr(ρ2A ) and tr(ρA ).
decomposed in the Pauli basis. Therefore, we ran 64 independent experiments to get each individual Pauli. For each Pauli term out of 64, we designed a specific circuit mainly according to its Pauli weight. Without loss of generality, let us take the circuits in Fig. S6 which can drive the system from the thermal equilibrium state to IZZIXX and ZZZXXX as examples. Each circuit can in fact be divided into three steps: (a) Eliminate all other 13 C signals but C4 . At room temperature, the thermal equilibrium state of this system is highly mixed, with the form ρeq =
6 1− I + Zi , 26 i=1
(3)
where ≈ 10−5 describes the polarization, and I is a 26 × 26
3 C1
C2
C3
C4
C5
C6
C-13 labeled 6-qubit system Dichloro-cyclobutanone
C1
8779
C2
57.58
30020
C3
0.30
0
10333
C4
37.48
1.25
29.02
11928
C5
-1.66
5.54
-3.53
34.57
C6
C3
C5
C4 C1
34381
C6
2.62
-1.25
33.16
21.75
33.16
15745
T1
3.61
7.99
3.72
3.64
7.80
9.89
T2
0.31
0.40
0.25
0.38
0.40
0.40
*
C2
FIG. S5. Molecular structure and Hamiltonian parameters of per-13 C labeled (1S,4S,5S)-7,7-dichloro-6-oxo-2-thiabicyclo[3.2.0]heptane4-carboxylic acid. The diagonal elements are the chemical shifts (in Hz), and the off-diagonal elements are the J-couplings between two spins (in Hz). The relaxation times T1 and T2∗ (in seconds) are also listed at bottom. All parameters are obtained on a Bruker DRX 700 MHz spectrometer at room temperature.
1/2J46
(a) C1
1/2J12
1/2J34
1/4J12
1/4J14
C2
1/4J12
1/4J12
C3
1/4J34
1/4J34
C4
1/4J34
1/4J46
C5
1/4J34
1/4J45
C6
1/4J46
(b)
1/2J46
C1
1/2J12
1/4J12
The final rank-6 PT state was obtained by the summation over all the above 64 Pauli terms.
D.
Six-qubit full state tomography
To verify the goodness of the prepared PT, we performed the six-qubit quantum state tomography. The density matrix was reconstructed through 110 readout pulses [5], where all of them are local. All the readout pulses were realized by 1 ms GRAPE pulses with fidelity over 99.5%. The state fidelity between the experimentally prepared PT and its theoretical form is about 85.0%, as shown in the main text.
E.
1/4J34
C4
(c) Evolve to the target Pauli term. Once we have the diagonal multi-coherence states, local π/2 rotations were applied on the target spins to generate the desired Pauli terms. For instance, in Fig. S6, we applied Ry5,6 (π/2) and Ry4,5,6 (π/2) on IZZIZZ and ZZZZZZ to generate the target Pauli IZZIXX and ZZZXXX, respectively.
1/4J12
1/4J14
C2 C3
rotations plus free evolutions of the internal Hamiltonian to drive the system from IIIZII to the diagonal multi-coherence state such as IZZIZZ and ZZZZZZ in Fig. S6. These single-qubit operations include π and π/2 rotations, and were optimized by the gradient ascent pulse engineering (GRAPE) technique [3, 4]. Each pulse has a 1 ms duration with the fidelity over 99.2%. Moreover, to minimize the discrepancy between the simulated shape and its real performance in experiment, each pulse was rectified via a feedback control technique by a pickup coil attached to the spectrometer.
Decoherence simulation
1/4J46
C5
1/4J45
C6
1/4J46
Ry (90s)
R y (90s)
Selective
Pulse
Gradient Pulse
FIG. S6. Pulse sequences that generate (a) IZZIXX and (b) ZZZXXX from the thermal equilibrium state, respectively. The sequence contains local rotations, free evolutions, and z-gradient pulses to destroy the unwanted coherent terms. The free evolution time 1/(2J) is in the magnitude of 10 ms, determined by the J-coupling strengths of the relevant spins.
identity matrix. We used a collective 1 ms π/2 pulse to rotate all 13 C spins except C4 to the transverse plane, and applied a z-gradient pulse consequently. The remained state now is IIIZII . (b) Generate multi-coherent state. We used single-qubit
To numerically simulate the decoherence effect in our sixqubit system, we made the following assumptions: the environment is Markovian; only the T2∗ dephasing mechanism is taken into account since T1 effect is negligible in our circuit; the dephasing noise is independent between all qubits; the dissipator and the total Hamiltonian commute in each pulse slice as the Δt = 10 μs is small. With these assumptions, we simplified and solved the master equation in two steps for each Δt: evolve the system by the propagator calculated by the internal and control pulse Hamiltonian, and subsequently apply the dephasing factors according to the coherent orders for Δt which is an exponential decay of the off-diagonal elements in the density matrix. For each experiment of the 64 runs, we simulated the above process and obtained the signal’s decay due to decoherence. From the experimental result, we then compensated for this decay, and a new state in which the decoherence effect was taken into account was thus achieved. The fidelity now between the rescaled experimental state and the theoretical PT is boosted to 93.7%.
4
[1] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Phys. Rev. Lett. 77, 198 (1996). [2] E. M. Rains, IEEE Trans. Inf. Theory 45, 266 (1999).
[3] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrggen, and S. J. Glaser, J. Magn. Reson. 172, 296 (2005). [4] C. A. Ryan, C. Negrevergne, M. Laforest, E. Knill, and R. Laflamme, Phys. Rev. A 78, 012328 (2008). [5] G. M. Leskowitz and L. J. Mueller, Phys. Rev. A 69, 052302 (2004).
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