Demonstration of Blind Quantum Computing

In the paper

“Here we show the first such experimental demonstration of blind quantum computation where the input, computation, and output all remain unknown to the computer.”

• Also stated in: § Experimental Demonstration of Blind Quantum Computing · p. 2 · boosted

  “We experimentally demonstrate the concept of BQC via a series of blind computations on four-qubit blind cluster states.”

In the paper

“Remarkably, the client only needs to be able to prepare and transmit individual photonic qubits.”

In the paper

“In fact, for the four-qubit blind cluster state it is sufficient for the client to be able to prepare only one or two of the qubits in arbitrary states | θ j 〉 for delegating various one- and two-qubit circuits as well as quantum algorithms, see Figure 2.”

In the paper

“We experimentally demonstrate the concept of BQC via a series of blind computations on four-qubit blind cluster states.”

In the paper

“Moreover, we have implemented 1962 different four-qubit measurements with 31392 measured outcomes.”

In the paper

“By aligning our setup to produce | Φ (2 ,n ) L 〉 for n = 0 , . . . , 7 and | Φ (6 ,m ) L 〉 for m = 0 , 4, we have demonstrated for the first time the preparation of various four-qubit blind cluster states.”

In the paper

“Our computed fidelities for the various blind cluster states achieve maximum values of up to 67 . 9 ± 0 . 4% via local unitary transformation.”

In the paper

“Here we demonstrate a blind implementation of Grover's search for n = 2, where blindness ensures that the server is unable to distinguish the actual computation from within a given family of circuits implementing ( I ⊗ R z ( ξ ) H ).”

In the paper

“By fixing θ 2 and varying θ 3 , we can experimentally demonstrate a blind X -rotation.”

In the paper

“In a similar way, the consistency with blindness of the rotated horseshoe cluster | Φ ˆ θ 〉 ⊃ (Figure 2D) can be shown.”

In the paper

“Using the tomographic measurements performed on these input states we determine χ of such states to be 0 . 169 ± 0 . 074, far below the three bits necessary to uniquely identify the client's choice of φ 2 and φ 3 , proving that within the assumptions of our model these experimental implementations of the protocol maintains close to perfect blindness.”

In the paper

“and 0 . 895 ± 0 . 022 for the balanced oracle”

In the paper

“Whereas previous realisations [10, 12] are not amenable to blind implementations, our computation, embedded into the blind triangle cluster | Φ ˆ θ 〉 △ (Figure 2F), remains blind.”

In the paper

“By varying θ 3 and averaging over all resulting density matrices, we obtain a totally mixed state with a linear entropy of 0 . 989 ± 0 . 010 that is close to the entropy of 1 for a perfectly mixed state (see Figure 3C).”

In the paper

“In the same way, a blind Z -rotation can be shown experimentally by using the four-qubit linear blind cluster state | Φ ̂ θ 〉 ← , which has the order of measurements going from qubit 4 down to qubit 1.”

In the paper

“Consistency with blindness can be seen by averaging over all output states, giving as a result a totally mixed state with a linear entropy of 0 . 955 ± 0 . 011.”

In the paper

“In contrast to previously realized implementations of Deutsch's algorithm using traditional cluster states [16, 46], we exploit blind staircase cluster states | Φ ˆ θ 〉 ⌟⌜ for the implementation of this quantum algorithm for the case n = 1.”

What the sources say

• Not checked yetcites [16] Vallone, G., Donati, G., Bruno, N., Chiuri, A. & Mataloni, P. Experimental realization of the Deutsch-Jozsa algorithm with a six-qubit cluster state. Phys. Rev. A 81 , 50302 (2010). source
• Not checked yetcites [46] Tame, M. S. et al. Experimental realization of Deutsch's algorithm in a one-way quantum computer. Phys. Rev. Lett. 98 , 140501 (2007). source

In the paper

“We also realize blind computations based on the blind staircase cluster | Φ ˆ θ 〉 ⌟⌜ (Figure 2E) and blind triangle cluster | Φ ˆ θ 〉 △ (Figure 2F).”

In the paper

“However even when this value is maximised over all possible prior distributions on the choice of states, it increases only slightly to 0 . 185 ± 0 . 087.”

In the paper

“Our demonstration uses a selection of four states which suffices to hide the choice of rotations among four possibilities: R z ( π/ 2 ± π ) ⊗ R z ( π/ 2 ± π ).”

In the paper

“and F = 0 . 887 ± 0 . 033 for the balanced oracle”

In the paper

“with the algorithm producing the correct result with probabilities 0 . 899 ± 0 . 006 for the constant”

In the paper

“One family of two-qubit gates generated in our experiment is based on the blind horseshoe cluster | Φ ˆ θ 〉 ⊂ , where measuring qubits 2 and 3 of the blind cluster state performs a transformation on the logical input qubits (Figure 2C).”

In the paper

“We note that due to the down-conversion process the client rather prepares arbitrarily rotated Bell pairs instead of single qubits”

In the paper

“Generating a family of four-qubit blind cluster states”

In the paper

“The different photon emissions then interfere at the PBSs with average visibilities of 0 . 85.”

In the paper

“These main error contributions, together with minor errors like polarisation drifts, decrease the fidelity of our blind cluster states with respect to the ideal state.”

In the paper

“Additional errors arise due to phase drifts during the measurements.”

In the paper

“In our experiment, the emitted Bell pairs show a typical visibility of about 0 . 9, depending on the specific experimental setting.”

In the paper

“The final output state is a superposition of all these four terms.”

In the paper

“Figure 6 shows relevant theoretical predictions as well as experimental outcomes which confirm the quantum nature of the server.”

In the paper

“we obtained a universal set of single-qubit and non-trivial two-qubit quantum logic gates”

In the paper

“This is underlined by an analysis of the data of Figure 6 (main paper) where we obtain a value of χ 2 of 1 . 6 when assuming Poissonian errors only.”

In the paper

“We have experimentally demonstrated the concept of blind quantum computing.”

In the paper

“We experimentally demonstrate the testing procedure using statistics of several outcomes for different measurement instructions.”

In the paper

“Including the errors mentioned above, we obtain a χ 2 value satisfactorily close to one (about 1 . 1).”

In the paper

“as well as implementations of Deutsch's and Grover's algorithms.”

In the paper

“By changing the phases of the entangled pairs, we can now manipulate the values of θ 2 and θ 3 of the client's qubits.”

In the paper

“Measuring qubit 1 in the eigenstates of σ x , σ y , or σ z has the effect of preparing the input on qubit 2 in the state | 0 〉 , | + i 〉 or | + 〉 , respectively, where | + i 〉 = 1 / √ 2( | 0 〉 + i | 1 〉 ).”

In the paper

“As shown in Figure 1, these photonic states can be combined via optical gates to create a universal resource state for BQC [29].”

What the sources say

• Not checked yetcites [29] Broadbent, A., Fitzsimons, J. & Kashefi, E. Universal blind quantum computation. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science , 517-526 (2009). source

In the paper

“Similarly, obtaining all the possible measurement outcomes is equivalent to implementing all possible values of r j , as if the client randomly re-interprets the measurement outcomes, implicitly subsuming r j .”

In the paper

“On these cluster states, adaptive single-qubit measurements alone are sufficient to implement deterministic universal quantum computation.”

In the paper

“Remarkably, the recent theoretical work by Broadbent, Fitzsimons, and Kashefi [29] overcomes this limitation and shows that quantum computers can provide unconditional security in data processing - a hitherto unrecognized potential of quantum computers that is not known to be achievable classically.”

What the sources say

• Not checked yetcites [29] Broadbent, A., Fitzsimons, J. & Kashefi, E. Universal blind quantum computation. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science , 517-526 (2009). source

In the paper

“As long as the angle θ 1 of the rotated qubit is unknown, the real rotation remains secret.”

In the paper

“However, the remarkably feature of the BQC protocol is that the client's privacy is always preserved, whether or not feed-forward mechanisms have been implemented.”

In the paper

“The four-qubit linear blind cluster | Φ ̂ θ 〉 → (Figure 2) can be used to implement an arbitrary single-qubit unitary gate.”

In the paper

“Measuring the first qubit, initially in state | θ 1 〉 , of a onedimensional linear blind cluster in the basis |± δ 1 〉 = 1 / √ 2( | 0 〉 ± e iδ 1 | 1 〉 ) has the effect of applying a singlequbit rotation R z ( -δ 1 + θ 1 ), on the encoded input state | + 〉 , followed by a Hadamard, H .”

In the paper

“Therefore, a quantum computation is hidden as long as these measurements are successfully hidden.”

In the paper

“Note that, whenever all qubits are measured in our setup, this method allows the client's choice of configuration to also be hidden from the server.”

In the paper

“Two-qubit gates are required for universal quantum computation; by choosing the order of measurements in a suitable way, the blind cluster | Φ ˆ θ 〉 implements blind two-qubit gates (Figure 2C-F).”

In the paper

“We are thus left with a three-qubit linear cluster state that implements a single-qubit rotation gate with rotations determined by the measurements of the second and third qubits; this rotates the input qubit | Ψ in 〉 to the final state | Ψ out 〉 = R x ( -φ 3 ) R z ( -φ 2 ) | Ψ in 〉 , where R x ( α ) = exp( -iασ x / 2).”

In the paper

“The state | Φ ˆ θ 〉 △ is obtained via local complementation [44] on qubit 2 of | Φ ˆ θ 〉 ⊂ .”

What the sources say

• Not checked yetcites [44] Van den Nest, M., Dehaene, J. & Moor, B. D. Graphical description of the action of local Clifford transformations on graph states. Phys. Rev. A 69 , 022316 (2004). source

In the paper

“Blindness of qubit 3 guarantees that the quantum server cannot distinguish between the execution of each of these scenarios (constant or balanced oracles) and corresponding families of quantum circuits.”

In the paper

“Whereas previous realisations [10, 12] are not amenable to blind implementations, our computation, embedded into the blind triangle cluster | Φ ˆ θ 〉 △ (Figure 2F), remains blind.”

What the sources say

• Not checked yetcites [10] Walther, P. et al. Experimental one-way quantum computing. Nature 434 , 169-176 (2005). source
• Not checked yetcites [12] Prevedel, R. et al. High-speed linear optics quantum computing using active feed-forward. Nature 445 , 65-69 (2007). source

In the paper

“However, blindness on qubit i is guaranteed only if the resulting measurement can be expressed as a basis |± δ ′ i 〉 .”

In the paper

“If it has no quantum technology in hand, it attempts to use its classical devices and guesses the outcome for the client's computation wrong with probability at least 1/8.”

In the paper

“and thus our constructions works for Deutsch's and Grover's algorithms.”

In the paper

“Thus the feedforward structure for φ 4 is given by the addition of an optional multiple of π . This is completely hidden by the usual addition of a random πr 4 , included in δ 4 .”

In the paper

“In fact, these indicate a lower bound for the actual error that takes all the experimental imperfections into account.”

In the paper

“Our optimised protocol guarantees full blindness of a computation, even though not all qubits have an a initial rotation.”

In the paper

“Here, we show that algorithms that admit a measurement pattern in which the secret can be encoded over a few qubits can be made fully blind using only a few blind qubits.”

In the paper

“In the setting of our experiment we demonstrate a notion of verification that can be used as a heuristic probabilistic test for whether the server indeed possesses any quantum technology or is a completely classical device.”

In the paper

“Better bounds can be achieved using statistics of several rounds by comparing it with the known theoretical statistics to test whether the quantum-computing server is producing the expected outcome or not.”

In the paper

“which enables a compact client-server network without affecting the blindness.”

In the paper

“The emission of only one entangled pair in the forward direction and only one pair in the backward direction results in two different four-photon terms: | H 〉 1 | H 〉 2 | H 〉 3 | H 〉 4 and -e i ( θ 2 + θ 3 ) | V 〉 1 | V 〉 2 | V 〉 3 | V 〉 4 due to the properties of the PBSs.”

In the paper

“Thus, from the server's point of view, the process of measuring qubit 4 is independent of φ 2 and φ 3 .”

In the paper

“The blind cluster state in the laboratory basis is composed of four terms that correspond to different emissions of four photons.”

In the paper

“By instructing the quantum server to measure, for example, this statistical distribution, the client can see if the outcomes coincide with the expectations.”

In the paper

“Hence, the quantum server, despite receiving classical information δ 2 and δ 3 and quantum states | θ 2 〉 and | θ 2 〉 , cannot distinguish between the possible choices of the client's measurements on qubits 2 and 3.”

In the paper

“This optimisation is due to the fact that for φ 4 ∈ { 0 , π/ 2 , π, 3 π/ 2 } , a sign flip on φ ( φ → -φ ) can be re-interpreted as the (possible) addition of π to φ ( φ → φ + π ).”

In the paper

“This construction works as long as measurements following the blind qubit measurements are chosen in the Clifford group (integer multiples of π/ 2)”