Blind quantum computation protocol in which Alice only makes measurements

Blind quantum computation protocol in which Alice only makes measurements

Abstract

Blind quantum computation is a new secure quantum computing protocol which enables Alice who does not have sufficient quantum technology to delegate her quantum computation to Bob who has a fully-fledged quantum computer in such a way that Bob cannot learn anything about Alice’s input, output, and algorithm. In previous protocols, Alice needs to have a device which generates quantum states, such as single-photon states. Here we propose another type of blind computing protocol where Alice does only measurements, such as the polarization measurements with a threshold detector. In several experimental setups, such as optical systems, the measurement of a state is much easier than the generation of a single-qubit state. Therefore our protocols ease Alice’s burden. Furthermore, the security of our protocol is based on the no-signaling principle, which is more fundamental than quantum physics. Finally, our protocols are device independent in the sense that Alice does not need to trust her measurement device in order to guarantee the security.

pacs:
03.67.-a

A first-generation quantum computer will be implemented in the “cloud” style, since only limited number of groups, such as governments and huge industries, will be able to possess it. How can a client of such a cloud quantum computing assure the security of his/her privacy? Protocols of blind quantum computation (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11) provide a solution. Blind quantum computation is a new secure quantum computing protocol which enables a client (Alice) who has only a classical computer or a primitive quantum device which is not sufficient for universal quantum computation to delegate her computation to a server (Bob) who has a fully-fledged quantum computer without leaking any Alice’s privacy (i.e., which algorithm Alice wants to run, which value Alice inputs, and what is the output of the computation) to Bob (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11).

The first example of blind quantum computation was proposed by Childs (1) where the quantum circuit model was used, and the register state was encrypted with quantum one-time pad (12) so that Bob who performs quantum gates learns nothing about information in the quantum register. In this method, however, Alice needs to have a quantum memory and the ability to perform the SWAP gate. The protocol proposed by Arrighi and Salvail (2) is that for the calculation of certain classical functions, i.e., not the universal quantum computation, and it requires Alice to prepare and measure multi-qubit entangled states. Furthermore, it is cheat-sensitive, i.e., Bob can gain information if he does not mind being caught. Finally, in their protocol, Bob knows the unitary which Alice wants to implement. Aharonov, Ben-Or and Eban’s protocol (4) requires a constant-sized quantum computer with a quantum memory for Alice.

On the other hand, in 2009, Broadbent, Fitzsimons and Kashefi (3) proposed a new blind quantum computation protocol which uses the one-way model (13); (14); (15); (16). In their protocol, all Alice needs are a classical computer and a primitive quantum device, which emits randomly rotated single-qubit states. In particular, Alice does not require any quantum memory and the protocol is unconditionally secure (i.e., Alice’s input, output, and algorithm are secret to Bob whatever Bob does). Recently, this protocol has been experimentally demonstrated in an optical system (7). Furthermore, this innovative protocol has inspired several new other protocols which can enjoy more robust blind quantum computation. In Ref. (5), two protocols which enable blind measurement-based quantum computation on the Affleck-Kennedy-Lieb-Tasaki (AKLT) state (17); (18) have been proposed. In Ref. (8), a protocol of the blind topological measurement-based quantum computation (19); (20); (21) has been proposed. Due to the topological protection, it is fault-tolerant (19); (20); (21). The error threshold of the blind topological model has been shown to be comparable to that of the original (20); (19) (i.e., non-blind) topological quantum computation (8).

Before starting the main part of this paper, let us quickly review the protocol of Ref. (3). In this protocol, Alice and Bob share a classical channel and a quantum channel. The protocol runs as follows: (1) Alice prepares randomly-rotated single-qubit states , where is a random angle, and sends them to Bob through the quantum channel. (2) Bob creates a certain two-dimensional graph state, which is called the brickwork state (3), by applying the C gates among . (3) Alice calculates the measurement angle on her classical computer, and sends it to Bob through the classical channel. (4) Bob performs the measurement in that angle, and returns the measurement result to Alice. (5) They repeat (3)-(4) until the computation is finished.

If Bob is honest, Alice obtains the correct answer of her desired quantum computation (3). Furthermore, it was shown that whatever evil Bob does, Bob cannot learn anything about Alice’s input, output, and algorithm (3).

The motivation of the blind quantum computation is to enable Alice, who does not have any sophisticated technology and enough knowledge, to perform universal quantum computation. Therefore, there are two important goals. One is to make Alice’s device as classical as possible, since Alice is not expected to have any expensive laboratory which can maintain the coherence of complicated quantum experimental setups. The other is to exempt Alice from the precise verification of her device, since Alice is not expected to have enough technology and knowledge to verify her device. Such a verification is important since she might buy the device from a company which is under the control of Bob, and therefore the device might not work as Alice expects. For example, if Alice is supposed to send a single-photon to Bob, Alice must confirm that more than two identical photons are not sent to Bob, since otherwise Bob might be able to gain some information by using, e.g., the photon-number-splitting (PNS) attack (22); (23); (24); (25), which is a well-known technique in quantum key distribution (QKD). In Ref. (6), a first step to the first goal, namely making Alice’s device more classical, was achieved. They proposed an ingenious protocol of the blind quantum computation in which what Alice needs to prepare are not single-photon states but coherent states. Since coherent states are considered to be more classical than single-photon states, their protocol allows Alice’s device to be more classical.

In this paper, we show that Alice who has only a measurement device can perform the blind quantum computation. In several experimental setups, such as quantum optical systems, the measurement of a state, e.g., the polarization measurement of photons with a threshold detector, is much easier than the generation of a single-qubit state, such as a single-photon state. Therefore, our results achieve the above mentioned first goal, namely making Alice’s device more classical. As we will see later, our protocols can cope with the particle loss in the quantum channel between Alice and Bob and the measurement inefficiencies of Alice’s measurement device. It also means that our protocols allow Alice’s device to be more classical.

We propose two new protocols, Protocol 1 and Protocol 2. Importantly, the security of Protocol 1 is based on the no-signaling principle (27), which is more fundamental than quantum physics (27). Therefore, even if Bob does a super-quantum (but no-signaling) attack, Alice’s privacy is still guaranteed. Furthermore, the device-independence (26) is attained for the security of Protocol 1. Hence the above second goal is also achieved. The security based on the no-signaling principle and device-independence are important subjects in quantum key distribution, and much researches have been done within the decade (28). However, Protocol 1 cannot cope with a high channel loss. Protocol 2 can tolerate any high channel loss, but the device-independence becomes weaker.

Protocol 1.— Our first protocol runs as follows: (1) Bob prepares a resource state of measurement-based quantum computation. Any resource state can be used for this purpose. For example, the two-dimensional cluster state (13); (14); (15), the three-dimensional cluster state for the topological quantum computation (19); (20); (21), the thermal equilibrium states of a nearest-neighbour two-body Hamiltonian with spin-2 and spin-3/2 particles (29) or solely with spin-3/2 particles (30) at a finite temperature for the topological measurement-based quantum computation, resource states for the quantum computational tensor network (31); (32); (33); (34); (35); (36); (37), the one-dimensional or two-dimensional AKLT states (18); (38); (39), the tri-cluster state (40), and states in the Haldane phase (41). (2) Bob sends a particle of the resource state to Alice through the quantum channel. (3) Alice measures the particle in a certain angle which is determined by the algorithm in her mind. They repeat (2)-(3) until the computation is finished.

Obviously, at the end of the computation, Alice obtains the correct answer of her desired quantum computation if Bob is honest, since what Alice and Bob did is nothing but a usual measurement-based quantum computation. (It is something like the following story: Alice and Bob are in the same laboratory. The preparation and the maintenance of the resource state, which are boring routines, are done by a student Bob, whereas the most exciting part of the measurement-based quantum computation, namely the measurements and the collection of data are done by his boss Alice. Somehow, there is no communication between the boss and the student.)

It is also easy to understand that whichever states evil Bob prepares instead of the correct resource state, and whichever states evil Bob sends to Alice, Bob cannot learn anything about Alice’s information, since Alice does not send any signal to Bob and therefore because of the no-signaling principle (27) Bob cannot gain any information about Alice by measuring his system (42): If Alice could transmit some information to Bob by measuring her system, it contradicts to the no-signaling principle (44). (Note that we assume there is no unwanted leakage of information from Alice’s laboratory. For example, Bob cannot bug Alice’s laboratory. It is the standard assumption in the quantum key distribution (45).) In Sec. I of Appendix, we give the mathematical proof of the security of Protocol 1 based on the no-signaling principle.

Protocol 1 has four advantages. First, unlike Ref. (3), no random-number generator is required for Alice. This is advantageous since it is not easy to generate completely random numbers, and the random-number generator might be provided by a company under the control of Bob. Second, the security of the protocol is device independent in the sense that Alice does not need to trust her measurement device in order to guarantee the security, since whatever Alice’s device does, Bob cannot gain any information about Alice’s computation (due to the no-signaling principle) as long as there is no unwanted leakage of information from Alice’s laboratory, which is the standard assumption in quantum key distribution (45). Third, the proof of the security is intuitive and very simple, and it is based on the no-signaling principle (27), which is more fundamental than quantum physics (27). (Even if quantum physics is violated in a future, Protocol 1 survives as long as the no-signaling principle holds.) Finally, any model of measurement-based quantum computation (such as the cluster model (13); (14); (15), the AKLT models (18); (38); (39), and the topological model (19); (20); (21), etc.) can be directly changed into a blind model: Bob has only to let Alice do measurements. (On the other hand, in Ref. (5), many complicated procedures are required to make the AKLT measurement-based quantum computation blind.) Since no modification is required to make a model blind, the advantage of a model is preserved when it is changed into a blind model. For example, an advantage of doing the measurement-based quantum computation on the AKLT states is that the quantum computation is protected by the energy gap of a physically natural Hamiltonian (18); (38); (39). If the AKLT model is used in Protocol 1, Bob who prepares and maintains the resource AKLT state can enjoy that advantage, i.e., Bob’s state is protected by the energy gap. This is also the case for the models of Refs. (29); (30): If these models are used in Protocol 1, Bob can enjoy the advantage of these models, i.e., Bob does not need to keep his state in the ground state; His state is allowed to be the equilibrium state at a finite temperature.

A disadvantage of Protocol 1 is that the quantum channel between Alice and Bob must not be too lossy. (Throughout this paper, “the channel loss” includes the detection inefficiency of Alice’s device, since the detection inefficiency behaves like the channel loss.) On the other hand, in the previous protocols (3); (5); (7); (8); (9) where Alice sends randomly rotated particles to Bob, the high loss rate of the quantum channel is not crucial, since if Bob does not receive a particle due to the loss in the quantum channel, Bob has only to ask Alice to again generate and send another state with another random angle. One way of overcoming that disadvantage of Protocol 1 is to use a model which can cope with the particle loss. For example, it was shown in Ref. (46) that the topological measurement-based quantum computation (19); (20); (21) can cope with the heralded particle loss if the loss probability is below the threshold. If Bob uses this model, Alice and Bob can perform Protocol 1 without suffering from the particle loss as long as the loss rate of the quantum channel between Alice and Bob (and that of Alice’s device) is below the loss threshold calculated in Ref. (46).

Protocol 2.— If we want to have a protocol which is tolerant against any high channel loss rate, we have to give up the perfect no-singling, since Alice has to send some message to Bob when a particle is lost.

One way of making Protocol 1 tolerant against any high channel loss is to use the quantum teleportation. Let us consider the following protocol: (1) Bob prepares a resource state. (2) He creates a Bell pair, and sends a half of it to Alice. (3) If the particle is lost, Alice asks Bob to send it again. If Alice receives the particle, Alice lets Bob know it. (4) Bob teleports a particle of his resource state to Alice by using the Bell pair. (5) Bob sends the measurement result of the teleportation to Alice. (6) Alice measures the teleported particle in an angle which is determined by her algorithm (and Bob’s teleportation result).

This protocol is a modified version of Protocol 1 where Bob teleports a particle of his resource state instead of directly sending it to Alice. This protocol is loss tolerant, since if a half of a Bell pair is lost in the channel, Bob has only to send it again. However, this protocol has a huge problem: Alice has to have a single-particle quantum memory, since Alice’s measurement must be done after Bob’s teleportation (otherwise Alice cannot correct byproducts created by Bob’s teleportation). Such a quantum memory does not need to have a long coherence time since the quantum teleportation can be done quickly, but still the requirement of a quantum memory is disadvantageous to Alice.

Here, we introduce Protocol 2, which can avoid such a quantum memory. This is our second main result of this paper. The basic idea of Protocol 2 is that Alice “prepares” rotated states which “encode” algorithm in Bob’s place, and Bob performs a layer-by-layer measurement-based quantum computation with these rotated states. Protocol 2 runs as follows: (1) Bob creates a Bell pair, and sends a half of it to Alice through the quantum channel. (2) If Alice does not receive it, because of the channel loss, Alice asks Bob to send it again and goes back to (1). (3) If Alice receives the particle, she measures it in the basis , where is a certain angle (not a random angle) determined by the algorithm which Alice wants to run. ( for Clifford gates, and for a non-Clifford gate. Details will be explained in Sec. II of Appendix.) In this way, Alice can “prepare” a state which encodes the angle of the algorithm in Bob’s place. (4) Bob couples the half of the Bell pair which he has to a qubit of his register state by using the gate, and measures the qubit in the register state in the basis. This -basis measurement implements the quantum gate. (5) Bob sends the result of this -basis measurement to Alice through the classical channel. (6) They repeat (1)-(5) until the computation is finished.

In Sec. III of Appendix, we show the blindness of Protocol 2: whichever states evil Bob prepares and whichever states evil Bob sends to Alice, Bob cannot learn anything about Alice’s information.

One might think Alice’s measurement in step (3) has to be delayed until the end of step (5) so that she can feed-forward Bob’s measurement outcome as usual one-way model. However, this is not the case. In Protocol 2, by properly choosing the measurement basis, we give a way to postpone Alice’s feed-forwarding until her subsequent measurements, and hence she does not have to wait for Bob’s measurement outcome. This means that no quantum memory is required for Alice. In Sec. II of Appendix, we give a detailed explanations about how Alice should measure particles.

In this way, we can obtain a protocol which is loss tolerant. However, as we have mentioned earlier, there is a trade-off between loss tolerance and no-signaling. Protocol 2 is no longer no-signaling. In Protocol 1, no signal is transmitted from Alice to Bob, and therefore the no-signaling is completely satisfied. However, in Protocol 2, the message whether Alice receives a particle or not is sent from Alice to Bob, and therefore the no-signaling is no longer satisfied. One might think that such a message is not directly related to Alice’s measurement angles, and therefore the situation is “quasi” no-signaling. However, if we want to show the device-independence security, a special care is necessary. In Ref. (47) it was shown that in quantum key distribution if Alice and Bob use the same measuring device many times, some secret information can be broadcasted through the “legal” channel. Similar attack can be considered in our Protocol 2. For example, let us assume that Alice does the measurement in the angle . Then, the measuring device remembers the number 5, and pretends to loss the particle fifth times. Then Alice sends the message, “the particle is lost”, to Bob fifth times, and from that fact, Bob can know the number 5. One way of avoiding such an attack is, as is explained in Ref. (47), to discard the measuring device after using it, and to use new device for every measurement. The other way, which can be used in our Protocol 2, is that Alice generates a random bit when Bob sends a particle, and behaves as if the particle is lost (arrived) when (1). In this case, Alice needs a random number generator, but the evil measuring device can no longer do that attack.

Discussion.— In this paper, we have proposed protocols of blind quantum computation for Alice who does only measurements, such as the polarization measurement with a threshold detector. In quantum optics, for example, the state measurement is much easier than the single-qubit state generation. Therefore our scheme makes Alice more classical than the previous protocols (3); (5); (8) in certain experimental setups, such as optical systems. In the protocol of Ref. (6), Bob is required to perform the non-demolition photon-number measurement, which is not easy with the current technology. In our protocols, on the other hand, Bob is not required to have such an additional high technology.

We have proposed two protocols, Protocol 1 and Protocol 2. Procotol 1 is simple, its security is based on the no-signaling principle, and it satisfies the device-independent security. However, it can not tolerate a high channel loss rate. On the other hand, Protocol 2 can tolerate any high channel loss rate, although it is more complicated than Protocol 1 and no longer no-signaling. Appropriate one should be chosen depending on the situation.

Finally, let us briefly discuss about the verification (4); (3); (9); (48) of blind quantum computing. The verification is a way of Alice checking whether Bob is honestly following her protocol (4); (3); (9); (48). It is important for blind quantum computation, since evil Bob can just destroy the computation and Alice might accept wrong computational results. Methods of the verification for blind quantum computation of Ref. (3) were already proposed in Refs. (3); (9). Can we do the verification for our measuring Alice blind quantum computation? One simple way of doing verification is that Alice randomly chooses some subsystem of the resource state and measures the stabilizer operators in order to check whether Bob correctly creates the resource graph state. Recently, more efficient way of doing verification for our measuring Alice protocol has been proposed in Ref. (48), which uses the previous verification methods of Aharonov, Ben-Or, and Eban (4) and Fitzsimons and Kashefi (9). By using that verification method, Alice can check whether Bob is honestly doing computation or not.

TM was supported by JSPS. KF was supported by MEXT Grant-in-Aid for Scientific Research on Innovative Areas 20104003. We acknowledge Matty J. Hoban for bringing our attention to Ref. (47).

I Blindness of Protocol 1

We assume that the initial state of the computation is the standard state , and the preparation of the input state is included in the computational part. Therefore, we can assume without loss of generality that what Alice wants to hide are the computation angles of the measurement-based quantum computation and the final output of the computation. Intuitively, a protocol is blind if Bob, given all the classical and quantum information during the protocol, cannot learn anything about Alice’s computational angles and the output (3); (5); (6); (8).

Definition: In this paper, we call a protocol is blind if

  • The conditional probability distribution of Alice’s computational angles, given all the classical information Bob can obtain during the protocol, and given the measurement results of any POVMs which Bob may perform on his system at any stage of the protocol, is equal to the a priori probability distribution of Alice’s computational angles, and

  • The conditional probability distribution of the final output of Alice’s algorithm, given all the classical information Bob can obtain during the protocol, and given the measurement results of any POVMs which Bob may perform on his system at any stage of the protocol, is equal to the a priori probability distribution of the final output of Alice’s algorithm.

Intuitively, this means that Bob’s “certainty” about Alice’s information does not changed even if Bob does POVM on his system.

Theorem 1: Protocol 1 satisfies (B1).

Proof: Let be the random variable which represents Alice’s measurement angles, and be the random variable which represents the type of the POVM which Bob performs on his system. Let be the random variable which represents the result of Bob’s POVM. The two-party system is called no-signaling (27) from Alice to Bob iff

for all , , , and . Then,

This means that Bob cannot learn anything about Alice’s measurement angles.

Theorem 2: Protocol 1 satisfies (B2).

Proof: Let be the random variable which represents the output of Alice’s algorithm, and be the random variable which represents the type of the POVM which Bob performs on his system. Let be the random variable which represents the result of Bob’s POVM. Alice can change the output of her algorithm by changing the input. (For example, since what is implemented in the quantum computation is a unitary operation, two input states which are orthogonal with each other become two mutually-orthogonal output states.) Because of the no-signaling principle,

for all , , , and . Then,

Therefore, Bob cannot learn anything about the output of Alice’s algorithm.

Ii Correctness of Protocol 2

Protocol 2 runs as follows: (1) Bob prepares the Bell pair and sends the half of it to Alice. (2) If Alice does not receive it, she asks Bob to try again, and goes back to (1). (3) If Alice receives the particle, she does the measurement in the

basis. How to choose will be explained later. After her measurement, Bob has the state

where , is Alice’s measurement result, and

Note that . (4) Bob creates the state

where is the C gate between the first and the second qubits and is any Bob’s state. (5) Bob performs the measurement in the basis on the first qubit. Since commutes with , Bob obtains

(1)

if the measurement result is , where is the Hadamard gate.

For , Eq. (1) becomes

For , Eq. (1) becomes

where .

For , Eq. (1) becomes

where .

Note that

where is a Pauli byproduct. (Be careful that different Pauli byproducts are represented by the same character for simplicity.) This means that the operations

can be done deterministically (up to Pauli byproducts) if Alice and Bob repeat the above (1)-(5) three times. Therefore, if we consider the unit cell (Fig. 1), the operations

can be implemented deterministically up to some Pauli byproducts as is shown in Fig. 2. Note that this set is universal set, since

where is or , and is or . As is shown in Fig. 3, the unit cell can be tiled to create the universal two-dimensional graph state which resembles the brickwork state (3).

Figure 1: (Color online.) The unit cell for Protocol 2.
Figure 2: (Color online.) Operations which can be implemented in the unit cell. Red triangles are Pauli byproducts. In (c), the blue can be replaced with .
Figure 3: (Color online.) Tiling for Protocol 2.

Iii Blindness of Protocol 2

Theorem 3: Protocol 2 satisfies (B1).

Proof: Let be the random variable which represents the type of the POVM which Bob performs on his system, and be the random variable which represents the result of the POVM. Let be the random variable which represents Alice’s message to Bob about the channel loss. Let be the random variable which represents Alice’s measurement angles. Bob’s knowledge about Alice’s measurement angles is given by the conditional probability distribution of given , and :

From Bayes’ theorem, we have

Theorem 4: Protocol 2 satisfies (B2).

Proof: Let be the random variable which represents the type of the POVM which Bob performs on his system, and be the random variable which represents the the result of the POVM. Let be the random variable which represents Alice’s message to Bob about the channel loss. Let be the random variable which represents the output of Alice’s algorithm. Bob’s knowledge about the output of Alice’s algorithm is given by the conditional probability distribution of given , , and :

From Bayes’ theorem, we have

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