Demonstration of Blind Quantum Computing

The paper provides the first experimental demonstration of blind quantum computation in which the input, computation, and output all remain unknown to the computer.

“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 demonstrated blind quantum computation, the client only needs to be able to prepare and transmit individual photonic qubits.

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

For the four-qubit blind cluster state, it is sufficient for the client to prepare only one or two qubits in arbitrary states |θj〉 to delegate various one- and two-qubit circuits and quantum algorithms.

“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.”

The paper experimentally demonstrates the concept of BQC via a series of blind computations on four-qubit blind cluster states.

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

The paper implemented 1962 different four-qubit measurements with 31392 measured outcomes.

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

The paper has demonstrated for the first time the preparation of various four-qubit blind cluster states.

“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.”

The computed fidelities for the various blind cluster states achieve maximum values of up to 67.9 ± 0.4% via local unitary transformation.

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

The paper demonstrates a blind implementation of Grover's search for n=2, where blindness ensures the server is unable to distinguish the actual computation from within a given family of circuits implementing (I ⊗ Rz(ξ)H).

“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 ).”

By fixing θ2 and varying θ3, the paper experimentally demonstrates a blind X-rotation.

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

The consistency with blindness of the rotated horseshoe cluster |Φ̂θ〉⊃ can be shown.

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

Using tomographic measurements on the input states, the Holevo information χ is determined to be 0.169 ± 0.074.

“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.”

The algorithm produced the correct result with probability 0.895 ± 0.022 for the balanced oracle.

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

The paper's Grover computation, embedded into the blind triangle cluster |Φ̂θ〉△, remains blind.

“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.”

By varying θ3 and averaging over all resulting density matrices, the paper obtains a totally mixed state with a linear entropy of 0.989 ± 0.010, close to the entropy of 1 for a perfectly mixed state.

“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).”

A blind Z-rotation can be shown experimentally using the four-qubit linear blind cluster state |Φ̂θ〉←.

“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.”

Averaging over all output states for the blind horseshoe cluster gives a totally mixed state with a linear entropy of 0.955 ± 0.011.

“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 contrast to previously realized implementations of Deutsch's algorithm using traditional cluster states, the paper exploits blind staircase cluster states |Φ̂θ〉⌟⌜ for the implementation of this quantum algorithm for n=1.

“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.”

• cites [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). sourceNot checked yet

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• cites [46] Tame, M. S. et al. Experimental realization of Deutsch's algorithm in a one-way quantum computer. Phys. Rev. Lett. 98 , 140501 (2007). sourceNot checked yet

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The paper realizes blind computations based on the blind staircase cluster |Φ̂θ〉⌟⌜ and blind triangle cluster |Φ̂θ〉△.

“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).”

Even when χ is maximised over all possible prior distributions on the choice of states, it increases only slightly to 0.185 ± 0.087.

“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.”

The demonstration uses a selection of four states which suffices to hide the choice of rotations among four possibilities: Rz(π/2 ± π) ⊗ Rz(π/2 ± π).

“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 ).”

The obtained fidelity for the output state is F = 0.887 ± 0.033 for the balanced oracle.

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

The algorithm produced the correct result with probability 0.899 ± 0.006 for the constant oracle.

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

One family of two-qubit gates generated in the experiment is based on the blind horseshoe cluster |Φ̂θ〉⊂, where measuring qubits 2 and 3 performs a transformation on the logical input qubits.

“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).”

Due to the down-conversion process, the client prepares arbitrarily rotated Bell pairs instead of single qubits.

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

The experiment generated a family of four-qubit blind cluster states.

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

The different photon emissions interfere at the PBSs with average visibilities of 0.85.

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

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

“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.”

Additional errors arise due to phase drifts during the measurements.

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

In the experiment, the emitted Bell pairs show a typical visibility of about 0.9, depending on the specific experimental setting.

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

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

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

Figure 6 shows experimental outcomes that confirm the quantum nature of the server.

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

The experiment obtained a universal set of single-qubit and non-trivial two-qubit quantum logic gates.

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

An analysis of the data of Figure 6 yields a value of χ2 of 1.6 when assuming Poissonian errors only.

“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.”

• AssumptionThe authors state this as an assumption: In the calculations, Poissonian errors are always assumed.

The authors have experimentally demonstrated the concept of blind quantum computing.

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

The authors experimentally demonstrate the testing procedure using statistics of several outcomes for different measurement instructions.

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

Including the errors mentioned above, the obtained χ2 value is satisfactorily close to one (about 1.1).

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

The experiment implemented Deutsch's and Grover's algorithms.

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

By changing the phases of the entangled pairs, the values of θ2 and θ3 of the client's qubits can be manipulated.

“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.”

Measuring qubit 1 in the eigenstates of σx, σy, or σz prepares the input on qubit 2 in the state |0〉, |+i〉 or |+〉, respectively.

“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 〉 ).”

These photonic states can be combined via optical gates to create a universal resource state for BQC.

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

• cites [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). sourceNot checked yet

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Obtaining all possible measurement outcomes is equivalent to implementing all possible values of rj, as if the client randomly re-interprets the measurement outcomes.

“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 .”

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

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

Quantum computers can provide unconditional security in data processing, a potential not known to be achievable classically.

“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.”

• cites [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). sourceNot checked yet

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As long as the angle θ1 of the rotated qubit is unknown, the real rotation remains secret.

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

The client's privacy is always preserved whether or not feed-forward mechanisms have been implemented.

“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.”

The four-qubit linear blind cluster |Φ̂θ〉→ can be used to implement an arbitrary single-qubit unitary gate.

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

Measuring the first qubit of a one-dimensional linear blind cluster in the basis |±δ1〉 applies a single-qubit rotation Rz(-δ1+θ1) on the encoded input state |+〉 followed by a Hadamard.

“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 .”

A quantum computation is hidden as long as the measurements are successfully hidden.

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

Whenever all qubits are measured in the setup, the method allows the client's choice of configuration to also be hidden from the server.

“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.”

By choosing the order of measurements in a suitable way, the blind cluster |Φ̂θ〉 implements blind two-qubit gates.

“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).”

The remaining three-qubit linear cluster state implements a single-qubit rotation gate with rotations determined by measurements of the second and third qubits, rotating input |Ψin〉 to |Ψout〉 = Rx(-φ3)Rz(-φ2)|Ψin〉.

“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).”

The state |Φ̂θ〉△ is obtained via local complementation on qubit 2 of |Φ̂θ〉⊂.

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

• cites [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). sourceNot checked yet

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Blindness of qubit 3 guarantees that the quantum server cannot distinguish between the execution of the constant and balanced oracle scenarios and their corresponding families of quantum circuits.

“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.”

Previous realisations of Grover's algorithm are not amenable to blind implementations.

“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.”

• cites [10] Walther, P. et al. Experimental one-way quantum computing. Nature 434 , 169-176 (2005). sourceNot checked yet

  Not checked in the free run.

• cites [12] Prevedel, R. et al. High-speed linear optics quantum computing using active feed-forward. Nature 445 , 65-69 (2007). sourceNot checked yet

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Blindness on qubit i is guaranteed only if the resulting measurement can be expressed as a basis |±δ′i〉.

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

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

“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.”

The construction works for Deutsch's and Grover's algorithms.

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

The feedforward structure for φ4 is given by the addition of an optional multiple of π, which is completely hidden by the usual addition of a random πr4 included in δ4.

“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 .”

Poissonian errors indicate a lower bound for the actual error that takes all experimental imperfections into account.

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

• AssumptionThe authors state this as an assumption: In the calculations, Poissonian errors are always assumed.

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

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

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.

“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 setting of this experiment, the authors demonstrate a notion of verification that can be used as a heuristic probabilistic test for whether the server possesses any quantum technology or is a completely classical device.

“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.”

Better bounds can be achieved using statistics of several rounds by comparing them with known theoretical statistics to test whether the quantum-computing server is producing the expected outcome.

“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.”

This enables a compact client-server network without affecting the blindness.

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

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(θ2+θ3)}|V〉1|V〉2|V〉3|V〉4 due to the properties of the PBSs.

“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.”

From the server's point of view, the process of measuring qubit 4 is independent of φ2 and φ3.

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

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

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

By instructing the quantum server to measure a statistical distribution, the client can see if the outcomes coincide with the expectations.

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

The quantum server, despite receiving classical information δ2 and δ3 and quantum states |θ2〉 and |θ3〉, cannot distinguish between the possible choices of the client's measurements on qubits 2 and 3.

“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.”

For φ4 in {0, π/2, π, 3π/2}, a sign flip on φ can be re-interpreted as the possible addition of π to φ.

“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 ).”

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

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