Silicon-photonic quantum chip
The chip is fabricated by standard complementary metal-oxide-semiconductor processes. The waveguide circuit patterns are defined on an 8 inches silicon-on-insulator (SOI) wafer through the 248 nm deep ultraviolet (DUV) photolithography processes and the inductively coupled plasma (ICP) etching processes. Once the waveguides layer is fabricated, a layer of silicon dioxide (SiO2) of 1m thickness was deposited by plasma-enhanced chemical vapor deposition (PECVD). Finally, thermal-optical phase-shifters are patterned by a layer of 50-nm-thick titanium nitride (TiN) deposited on top of waveguides. Single photons were generated and guided in silicon waveguides with a cross-section of 450nm220nm. The photon-pair sources were designed with a length of 1.2cm. Multimode interferometers (MMIs) with a width of 2.8m and length of 27m were used as balanced beamsplitters. The chip was wired-bounded on a PCB and each phase-shifter was individually controlled by an electronic driver. An optical microscopy image of the chip is shown in Fig.2a.
In our experiment, we used a tunable continuous wave (CW) laser at the wavelength of 1550.12 nm to pump the nonlinear sources, which was amplified to 100mW power using an erbium-doped fiber amplifier (EDFA). Photon-pairs of different frequencies were generated in integrated sources by the spontaneous four wave mixing (SFWM) process, and then spatially separated by on-chip asymmetric Mach-Zehnder interferometers (MZIs). The signal photon was chosen at the wavelength of 1545.32nm and the idler photon at 1554.94nm. Single-photons were routed off-chip for detection by an array of fiber-coupled superconducting nanowire single-photon detectors (SNSPDs) with an averaged efficiency of 85%, and photon coincidence counts were recorded by a multichannel time interval analyzer (TIA). The rate of photons is dependent on the choice of projective measurement bases. In the typical setting of our experiments, for example, when the state is projected to the eigenbasis, the two-photon coincidence rate was measured to be ~kHz, and the integration time in the projective measurement was chosen as 5s.
Our quantum photonic chip is shown in Fig.2a, which integrates more than 400 photonic components, allowing arbitrary on-chip preparation, operation, and measurement of four-qubit hypergraph states. Key ability includes the multiqubit-controlled unitary operations CmU, where U represents the arbitrary unitary operation (e.g., U=Z in our experiment) and m is the number of control qubits. The realization of multi-qubit CmU gates relies on the transformation from the entanglement sources to the entangling operations, by using the process of entanglement generationspace expansionlocal operationcoherent compression"28.
Firstly, the four-dimensional Bell state is created by coherently exciting an array of four spontaneous four-wave mixing (SFWM) sources. A pair of photons with different frequencies are then separated by on-chip asymmetric Mech-Zehnder interferometers and routed to different paths, resulting in the four-dimensional Bell state29:
$${leftvert {{{{{{{rm{Bell}}}}}}}}rightrangle }_{4}=frac{{leftvert 0rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{s}{leftvert 0rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{i}+{leftvert 1rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{s}{leftvert 1rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{i}+{leftvert 2rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{s}{leftvert 2rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{i}+{leftvert 3rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{s}{leftvert 3rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}^{i}}{2},$$
(3)
where (leftvert krightrangle) (k=0,1,2,3) represents the logical bases of qudits, and the superscripts of s,i represent the signal and idler single-photon, respectively. The two-qubit states are mapped to the four-dimensional qudit state in both of the signal and idler single-photon as the following:
$$left{begin{array}{c}leftvert 00rightrangle_{{{{{{{{rm{qubit}}}}}}}}}to leftvert 0rightrangle_{{{{{{{{rm{qudit}}}}}}}}}hfill\ leftvert 01rightrangle_{{{{{{{{rm{qubit}}}}}}}}}to leftvert 1rightrangle_{{{{{{{{rm{qudit}}}}}}}}}hfill\ leftvert 10rightrangle_{{{{{{{{rm{qubit}}}}}}}}}to leftvert 2rightrangle_{{{{{{{{rm{qudit}}}}}}}}}hfill\ {leftvert 11rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}to {leftvert 3rightrangle }_{{{{{{{{rm{qudit}}}}}}}}}hfillend{array}right.$$
(4)
This results in the four-qubit state as:
$$leftvert Phi rightrangle= frac{{leftvert 00rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{s}{leftvert 00rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{i}+{leftvert 01rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{s}{leftvert 01rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{i}}{2}\ +frac{{leftvert 10rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{s}{leftvert 10rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{i}+{leftvert 11rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{s}{leftvert 11rightrangle }_{{{{{{{{rm{qubit}}}}}}}}}^{i}}{2},$$
(5)
where (leftvert krightrangle) (k=0,1) represents the logical bases of qubits. For clarity, we omit the subscript of qubit in the following.
Secondly, we expand the Hilbert space of the idler-photonic qubit into a 4-dimensional space. After the space expansion process, we add two ancillary qubits ({leftvert phi rightrangle }^{i}) (third ququart) into the state:
$${leftvert Phi rightrangle }_{1}=frac{{leftvert 00rightrangle }^{s}{leftvert 00rightrangle }^{i}{leftvert phi rightrangle }^{i}+{leftvert 01rightrangle }^{s}{leftvert 01rightrangle }^{i}{leftvert phi rightrangle }^{i}+{leftvert 10rightrangle }^{s}{leftvert 10rightrangle }^{i}{leftvert phi rightrangle }^{i}+{leftvert 11rightrangle }^{s}{leftvert 11rightrangle }^{i}{leftvert phi rightrangle }^{i}}{2}.$$
(6)
Thirdly, the ancillary two-qubit ({leftvert phi rightrangle }^{i}) are locally operated using arbitrary two-qubit unitary gates represented by Uij. We apply different unitary operations U00, U01, U10, and U11 on the ({leftvert phi rightrangle }^{i}) (marked by different colors in Fig.2a). This returns a state:
$${leftvert Phi rightrangle }_{2}= frac{{leftvert 00rightrangle }^{s}{leftvert 00rightrangle }^{i}{leftvert {phi }_{R}rightrangle }^{i}+{leftvert 01rightrangle }^{s}{leftvert 01rightrangle }^{i}{leftvert {phi }_{Y}rightrangle }^{i}}{2}\ +frac{{leftvert 10rightrangle }^{s}{leftvert 10rightrangle }^{i}{leftvert {phi }_{G}rightrangle }^{i}+{leftvert 11rightrangle }_{1}{leftvert 11rightrangle }^{s}{leftvert {phi }_{B}rightrangle }^{i}}{2},$$
(7)
where subscripts of {R(ed), Y(ellow), G(reen), B(lue)} represent the state after Uij. The Uij are realized by universal linear-optical circuits30.
Finally, to preserve quantum coherence, the which-process information is erased in the coherent compression process. This swaps the state information of the idler qubits as:
$${leftvert Phi rightrangle }_{3}= frac{{leftvert 00rightrangle }^{s}{leftvert {phi }_{R}rightrangle }^{i}{leftvert 00rightrangle }^{i}+{leftvert 01rightrangle }^{s}{leftvert {phi }_{Y}rightrangle }^{i}{leftvert 01rightrangle }^{i}}{2}\ +frac{{leftvert 10rightrangle }^{s}{leftvert {phi }_{G}rightrangle }^{i}{leftvert 10rightrangle }^{i}+{leftvert 11rightrangle }^{s}{leftvert {phi }_{B}rightrangle }^{i}{leftvert 11rightrangle }^{i}}{2},$$
(8)
Through the post-selection procedure of projecting the last two qubits into the superposition state ((leftvert 00rightrangle+leftvert 01rightrangle+leftvert 10rightrangle+leftvert 11rightrangle )/2), we coherently compress the 16-dimensional space back into the 4-dimensional space with a success probability of 1/4, and we obtain:
$${leftvert Phi rightrangle }_{4}=frac{{leftvert 00rightrangle }^{s}{leftvert {phi }_{R}rightrangle }^{i}+{leftvert 01rightrangle }^{s}{leftvert {phi }_{Y}rightrangle }^{i}+{leftvert 10rightrangle }^{s}{leftvert {phi }_{G}rightrangle }^{i}+{leftvert 11rightrangle }^{s}{leftvert {phi }_{B}rightrangle }^{i}}{2}.$$
(9)
In short, the process of entanglement generation-space expansion-local operation-coherent compression" results in the multi-qubit entangling gate as:
$$leftvert 00rightrangle leftlangle 00rightvert {U}_{00}+leftvert 01rightrangle leftlangle 01rightvert {U}_{01}+leftvert 10rightrangle leftlangle 10rightvert {U}_{10}+leftvert 11rightrangle leftlangle 11rightvert {U}_{11}.$$
(10)
By reprogramming the linear-optical circuits for local unitary operations Uij, we can realize different multi-qubits controlled unitary gates such as CmZ, m3. For example, the triple-controlled CCCZ gate can be obtained by setting the configuration as U00=U01=U10=II and U11=CZ. The quantum chip thus enables the generation, operation and measurement of arbitrary four-qubit hypergraph states.
We here adopt the method proposed in ref. 31 to characterize the CCCZ gate. Since the CCCZ gate is invariant with respect to the permutation of the controlled and target qubits, we can characterize the gate by measuring the input-output truth tables for four complementary product bases. In these bases, three of the qubits are prepared and measured in the computational basis states {(leftvert 0rightrangle,leftvert 1rightrangle)} while the fourth qubit is prepared and measured in the Hadamard basis states {(leftvert+rightrangle,leftvert -rightrangle)}. Inputting the product state (vert {psi }_{i,j}rangle) returns a product state of (vert {psi }_{i,j}^{{{{{{{{rm{(out)}}}}}}}}}rangle={U}_{CCCZ}vert {psi }_{i,j}rangle). The measured truth tables are shown in Fig.2. We define the average statistic classical state fidelity as ({{{{{{{{rm{F}}}}}}}}}_{{{{{{{{rm{c}}}}}}}}(j)}=mathop{sum }nolimits_{i=1,k=1}^{16}{p}_{ik}{q}_{ik}/16), where pik and qik are the theoretical and measured distribution. According to the Choi-Jamiolkowski isomorphism, we define the Choi matrix of an ideal CCCZ gate as 0, and the experimental Choi matrix as , from which the quantum process fidelity for the CCCZ gate can be written as ({{{{{{{{rm{F}}}}}}}}}_{chi }={{{{{{{rm{Tr}}}}}}}}[chi {chi }_{0}]/({{{{{{{rm{Tr}}}}}}}}[{chi }_{0}]{{{{{{{rm{Tr}}}}}}}}[chi ])), where ({{{{{{{rm{Tr}}}}}}}}[{chi }_{0}]=16) accounts for the normalization. We obtain the generalized Hodmann bound of fidelity31 (the lower bounded process fidelity) for the CCCZ gate, which can be estimated from the four above averaged state fidelities as FFc1+Fc2+Fc3+Fc44.
In this part, we show the rule of LU transformation when applying local Pauli operations on the hypergraph states of (leftvert {{{{{{{rm{HG}}}}}}}}rightrangle=({prod }_{ein E}{C}_{e}){leftvert+rightrangle }^{otimes n})9, where e is a hyperedge connecting vertices {i1,i2,...,im} and ({C}_{e}=I-2({leftvert 1rightrangle }_{{i}_{1}}{leftvert 1rightrangle }_{{i}_{2}}cdots {leftvert 1rightrangle }_{{i}_{m}})cdot ({leftlangle 1rightvert }_{{i}_{1}}{leftlangle 1rightvert }_{{i}_{2}}cdots {leftlangle 1rightvert }_{{i}_{m}})) is the corresponding multiqubit controlled-Z gates. To show the LU transformation, as an example, we consider the case when applying the Pauli X-operation on the kth qubit. The state can be written as:
$${X}_{k}leftvert HGrightrangle= {X}_{k}(mathop{prod}limits_{ein E}{C}_{e}){leftvert+rightrangle }^{otimes n}\= (mathop{prod}limits_{ein E,enotni k}{C}_{e}){X}_{k}(mathop{prod}limits_{ein E,eni k}{C}_{e}){leftvert+rightrangle }^{otimes n}\= (mathop{prod}limits_{ein E,enotni k}{C}_{e})cdot left[right.{X}_{k}(mathop{prod}limits_{ein E,eni k}{C}_{e}){X}_{k}left]right.{leftvert+rightrangle }^{otimes n}\= (mathop{prod }limits_{ein E,enotni k}{C}_{e})cdot (mathop{prod}limits_{ein E,eni k}{X}_{k}{C}_{e}{X}_{k}){leftvert+rightrangle }^{otimes n}.$$
(11)
Now we focus on to the single operator XkCeXk. Assume the edge e connects vertices {1,2,...,m} and for simplicity we can assume k=1 is the first vertex (this does not sacrifice generality). Following the above assumption, we can write the operator explicitly as:
$${X}_{k}{C}_{e}{X}_{k}= {X}_{k}(I-2leftvert 11cdots 1rightrangle leftlangle 11cdots 1rightvert ){X}_{k}\= I-2leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightvert$$
(12)
Next step we separate Ce out on the left side. Notice that (I={C}_{e}^{2}) and
$$leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightvert= (I-2leftvert 11cdots 1rightrangle leftlangle 11cdots 1rightvert )cdot leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightvert \= {C}_{e}leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightvert$$
(13)
Therefore, we have
$${X}_{k}{C}_{e}{X}_{k}= I-2leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightverthfill \ = {C}_{e}cdot ({C}_{e}-2leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightvert )hfill\ = {C}_{e}cdot (I-2leftvert 11cdots 1rightrangle leftlangle 11cdots 1rightvert -2leftvert 01cdots 1rightrangle leftlangle 01cdots 1rightvert )\ = {C}_{e}cdot left(I-2cdot underbrace{(leftvert 1rightrangle leftlangle 1rightvert+leftvert 0rightrangle leftlangle 0rightvert )}_{begin{array}{c}{I}_{k}end{array}}otimes underbrace{leftvert 1cdots 1rightrangle leftlangle 11cdots 1rightvert }_{begin{array}{c}m-1end{array}}right)hfill\= {C}_{e}({I}_{k}otimes {C}_{e/{k}})$$
(14)
where Ce/{k} represents the multiqubit controlled gates corresponding to a new hyperedge {1,2,...,k1,k+1,..,m}.
Finally, we complete the proof by substituting the above formula into Eq.(11), which leads to
$${X}_{k}leftvert Grightrangle= (mathop{prod}limits_{ein E,enotni k}{C}_{e})cdot (mathop{prod}limits_{ein E,eni k}{X}_{k}{C}_{e}{X}_{k}){leftvert+rightrangle }^{otimes n}\= (mathop{prod}limits_{ein E,enotni k}{C}_{e})cdot (mathop{prod}limits_{ein E,eni k}{C}_{e}({I}_{k}otimes {C}_{e/{k}})){leftvert+rightrangle }^{otimes n}\= (mathop{prod}limits_{ein E}{C}_{e})cdot (mathop{prod}limits_{ein E,eni k}{C}_{e/{k}}){leftvert+rightrangle }^{otimes n}.$$
(15)
Equation (15) shows the LU transformation rule: applying a local Pauli X gate on a qubit equals to applying a series of multiqubit controlled-Z gates which connect other qubits that share the same edge with it.
We take an example to illustrate local unitary transformation, as shown in Fig.1c. The initial state is
$$leftvert psi rightrangle= leftvert 0000rightrangle+leftvert 0001rightrangle+leftvert 0010rightrangle+leftvert 0011rightrangle \ +leftvert 0100rightrangle -leftvert 0101rightrangle+leftvert 0110rightrangle+leftvert 0111rightrangle \ +leftvert 1000rightrangle+leftvert 1001rightrangle+leftvert 1010rightrangle+leftvert 1011rightrangle \ -leftvert 1100rightrangle -leftvert 1101rightrangle -leftvert 1110rightrangle -leftvert 1111rightrangle$$
(16)
After applying X3, which flips the third qubit, the state becomes
$$leftvert psi rightrangle=leftvert 0000rightrangle+leftvert 0001rightrangle+leftvert 0010rightrangle+leftvert 0011rightrangle \+leftvert 0100rightrangle+leftvert 0101rightrangle+leftvert 0110rightrangle -leftvert 0111rightrangle \+leftvert 1000rightrangle+leftvert 1001rightrangle+leftvert 1010rightrangle+leftvert 1011rightrangle \ -leftvert 1100rightrangle -leftvert 1101rightrangle -leftvert 1110rightrangle -leftvert 1111rightrangle$$
(17)
which can be quickly verified as the expression for the second hypergraph state in Fig.1c. Following a similar procedure, the hypergraph can be simplified to only two edges as shown in Fig.1c. The rule of LU transformation can be graphically described as the X(k) operation on the qubit k removes or adds these hyper-edges in E(k) depending on whether they exist already or not, where E(k) represents all hyper-edges that contain qubit k but removing qubit k out. The Z(k) operation on the qubit k remove the one-edge on the qubit k.
We here derive the basis used for the evaluation of MK polynomials M4 and ({M}_{4}^{{prime} }). The general form of Mn is given as37:
$${M}_{n}=frac{1}{2}{M}_{n-1}({a}_{n}+{a}_{n}^{{prime} })+frac{1}{2}{M}_{n-1}^{{prime} }({a}_{n}-{a}_{n}^{{prime} })$$
(18)
where an and ({a}_{n}^{{prime} }) are single-qubit operators and M1=a1. ({M}_{n}^{{prime} }) can be obtained by interchanging the terms with and without the prime. In particular, for the four-qubit state, we then have M4 and ({M}_{4}^{{prime} }):
$$left{begin{array}{l}{M}_{4}=frac{1}{2}{M}_{3}({a}_{4}+{a}_{4}^{{prime} })+frac{1}{2}{M}_{3}^{{prime} }({a}_{4}-{a}_{4}^{{prime} })quad \ {M}_{4}^{{prime} }=frac{1}{2}{M}_{3}^{{prime} }({a}_{4}+{a}_{4}^{{prime} })-frac{1}{2}{M}_{3}({a}_{4}-{a}_{4}^{{prime} }).quad end{array}right.$$
(19)
Similarly, {M3,M2} and {({M}_{3}^{{prime} },{M}_{2}^{{prime} })} can be obtained. We instead use an alternative way by dividing the original 4-qubit operators into 2-qubit by 2-qubit parts because of the implementation of qubit-qudit mapping in our device. This leads to the construction of the MK polynomials M4 and ({M}_{4}^{{prime} }) from M2 and ({M}_{2}^{{prime} }):
$$left{begin{array}{l}{M}_{4}=frac{1}{2}left[right.{M}_{2}({a}_{3}{a}_{4}^{{prime} }+{a}_{3}^{{prime} }{a}_{4})+{M}_{2}^{{prime} }({a}_{3}{a}_{4}-{a}_{3}^{{prime} }{a}_{4}^{{prime} })left]right.quad \ {M}_{4}^{{prime} }=frac{1}{2}left[right.{M}_{2}^{{prime} }({a}_{3}{a}_{4}^{{prime} }+{a}_{3}^{{prime} }{a}_{4})-{M}_{2}({a}_{3}{a}_{4}-{a}_{3}^{{prime} }{a}_{4}^{{prime} })left]right..quad end{array}right.$$
(20)
In experiment, we first measured the ({M}_{2},, {M}_{2}^{{prime} },, ({a}_{3}{a}_{4}^{{prime} }+{a}_{3}^{{prime} }{a}_{4})) and (({a}_{3}{a}_{4}-{a}_{3}^{{prime} }{a}_{4}^{{prime} })), and then estimated the MK polynomials M4 and ({M}_{4}^{{prime} }). A total number of 64 bases are required for M4 and ({M}_{4}^{{prime} }), each of which is determined by the choice of the corresponding ai and ({a}_{i}^{{prime} }).
In blind quantum computation, clients use the expensive resource states shared by the server to perform their measurements. In such a scenario, the average fidelity of the states generated by the server has to be verified before computation. Ideally, the clients are capable of estimating a lower bound of the state fidelity and verifying genuine entanglement, without much cost. We here use a protocol of color-encoding stabilizers41. To achieve a verification of fidelity larger than 10, the number of states required is given by
$$N=leftlceil frac{{{{{{rm{ln}}}}}}(delta)}{{{{{{rm{ln}}}}}}(1-epsilon_0/s)} rightrceil,$$
(21)
where s is the minimum number of colors in the hypergraph state, is the significance level and 0 denotes the error. This formula can be better understood in the following form
$$delta ge {(1-{epsilon }_{0}/s)}^{N},$$
(22)
where the right-hand side represents a total passing probability of the total N tests for a state with an infidelity 0. When this probability is smaller than the chosen significance level and a passing event occurs on the client side, we can draw the conclusion that the real infidelity of the state generated from the server should satisfy <0 with a significance level .
A simple transformation of Eq. (21) gives
$$bar{F}ge scdot {delta }^{1/N}-(s-1).$$
(23)
In the ideal case, if the generated state is exactly the target hypergraph state, i.e, F=1, the probability of passing the test is always 100%, while increasing the number of tests will result in a tighter bound (smaller 0). In reality, for experimental states with non-unit fidelity, the total passing probability will decrease exponentially with the number of tests N. When we define the single-test passing probability as (bar{P}), the total passing probability will take the form of ({bar{P}}^{N}), which should be kept above the significance level . Therefore, for a selected significance level, the maximum number of tests, which corresponds to the tightest bound on fidelity, should satisfy ({bar{P}}^{N}=delta). Replacing by ({bar{P}}^{N}) in Eq. (23) thus returns
$$bar{F}ge scdot bar{P}-(s-1).$$
(24)
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