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Universal Superreplication of Unitary Gates G. Chiribella, Y. Yang, and C. Huang Center fo r Quantum Information, Institute fo r Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
(Received 3 December 2014; published 25 March 2015) Quantum states obey an asymptotic no-cloning theorem, stating that no deterministic machine can reliably replicate generic sequences of identically prepared pure states. In stark contrast, we show that generic sequences of unitary gates can be replicated deterministically at nearly quadratic rates, with an error vanishing on most inputs except for an exponentially small fraction. The result is not in contradiction with the no-cloning theorem, since the impossibility of deterministically transforming pure states into unitary gates prevents the application of the gate replication protocol to states. In addition to gate replication, we show that N parallel uses of a completely unknown unitary gate can be compressed into a single gate acting on 0(\og2N) qubits, leading to an exponential reduction of the amount of quantum communication needed to implement the gate remotely. DOI: 10.1103/PhysRevLett. 114.120504
PACS numbers: 03.67.Ac, 03.65.Aa
A striking feature of quantum theory is the impossibility of constructing a universal copy machine, which takes, as input, a quantum system in an arbitrary pure state and produces, as output, a number of exact replicas [1,2], Such an impossibility has major implications for quantum error correction [3,4] and cryptographic protocols such as key distribution [5,6], quantum secret sharing [7,8], and quantum money [9-12], The impossibility of universal copy machines is a hard fact: it equally affects deterministic [13-16] and probabi listic machines [17-19], whose performances coincide with those of deterministic machines when it comes to copying completely unknown pure states [18,19], A similar no-go result holds when the universal machine is presented a large number N of identical copies and is required to produce a larger number M > N of approximate replicas: if the replicas have nonvanishing overlap with the desired M - copy state, then the number of extra copies must be negligible compared to N [15], We refer to this fact as the asymptotic no-cloning theorem, expressing the fact that independent and identically distributed (I1D) sequences of pure states cannot be stretched by any significant amount. The asymptotic no-cloning theorem holds also for nonun iversal machines designed to copy continuous sets of states, provided that such machines work deterministically [19]. The impossibility of universal state cloning suggests similar results for quantum gates. Along this line, a no-go theorem for universal gate cloning was proven in Ref. [20], showing that no quantum network can perfectly simulate two uses of an unknown unitary gate by querying it only once. Optimal networks that approximate universal gate cloning were studied in Refs. [20,21], Very recently, Diir and coauthors [22] considered a nonuniversal setup designed to clone phase gates, i.e., gates generated by time evolution with a known Hamiltonian. In this scenario, they devised a quantum network that approximately simulates up to N 2 0031 -9007/15/114( 12)/120504(5)
uses of an unknown phase gate while using it only N times, with vanishing error in the large N limit. Such a result establishes the possibility of superreplication of phase gates— superreplication being the generation of M » N high-fidelity replicas from V » 1 input copies [19], Remarkably, superreplication of phase gates is achieved deterministically, whereas superreplication of phase states has an exponentially small probability of success. The main open question raised by Ref. [22] is whether deterministic superreplication occurs not only for phase gates, but also for arbitrary unitary gates. An affirmative answer would imply that the asymptotic no-cloning theorem only applies to states, whereas it is possible to stretch long IID sequences of reversible gates by up to a quadratic factor. In this Letter, we answer the question in the affirmative, establishing the possibility of universal superreplication of unitary gates, in stark contrast with the asymptotic no-cloning theorem for pure states. Given N uses of a completely unknown unitary gate U, we construct a quan tum network that simulates up to N 2 parallel uses of U, providing an output that is close to the ideal target for all possible input states except for an exponentially small fraction. The quadratic replication rate is optimal: every other network producing replicas at a rate higher than quadratic will necessarily spoil their quality, delivering an output that has vanishing overlap with the output of the desired gate. In addition to replication, we consider the task of gate compression, where the goal is to faithfully encode the action of a black box into a gate operating on a smaller quantum system. For a quantum system of dimension d, we show that N uses of a completely unknown gate can be encoded without any loss into a single gate acting only on (d - l )( d/ 2 + 1)log27V qubits, thus, allowing for an expo nential reduction of computational workspace. The number of qubits can be further cut down by nearly a half if one tolerates an error that vanishes on almost all inputs in the
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large N limit. The compression of IID gate sequences is the analogue of the compression of IID state sequences [23], recently demonstrated experimentally [24], Universal superreplication of qubit gates. —Let us start from the simple case of qubit gates, represented by unitary matrices in SU(2). A generic gate can be parametrized as Udn = exp{-i0n • j], where 9 G [0, 2k ) is a rotation angle, n = (nx,ny,nz) is a rotation axis, and j = {jxJyJz) is the vector of angular momentum operators (j, —oi/2,i=x,y,z). We define g: — (9. n) and label the unitary gate as Ug. Here, both 9 and n are completely unknown, differing from the setting of [22], where the rotation axis was fixed and only 9 was varying. In order to replicate unknown gates, we consider a network where N parallel uses of Ug are sandwiched between two quantum channels, C\ and C2, as in Fig. 1. The overall action of the network is described by the channel C2(U fN (g) T A)C\, with Ug{-) — Ug - Ufg and I A denoting the identity on a suitable ancillary system. To construct the channels C\ and C2, we decompose the Hilbert space of F qubits (K = N , M) into rotationally invariant subspaces. Choosing F to be even, we have K/2
( 1) 7=0
where j is the quantum number of the total angular momentum, 7Zj is a representation space, of dimension dj = 2j + 1, and M jk is a multiplicity subspace, of dimension mjK. The isomorphism in Eq. (1) is called the Schur transform and can be implemented efficiently in a quantum circuit [25,26]. We now introduce a cutoff on the quantum number j and define the subspace
n f ] = ®{Kj ® M jK).
(2 )
j 0 and to / = N /2 for M growing like N 2 or faster. For channel C2, we choose the inverse of Vj, namely,
where P\K1is the projector on H jK) and p0 is a fixed density matrix with support in '. The key observation is that most F-qubit states are left nearly unchanged by the (K)
channel £) , provided that K is large and J is large compared to \JK. Denoting by F^f'7 ' the fidelity between a generic F-partite pure state T) and its compressed version
S p m m , we have the following. Theorem 1. If |v&) is chosen uniformly at random, then, for every fixed e > 0, the probability that F p J] is smaller than 1 - e satisfies the bound
FIG. 1 (color online). Quantum network for gate replication. The network simulates M parallel uses of an unknown unitary gate Ug, while querying it only N times. The simulation is obtained by transforming the input state of M systems into the joint state of N systems plus an ancilla (via quantum channel C\), applying the unknown gate on the N systems, and then recom bining them with the ancilla via a quantum channel C2, which finally produces M output systems.
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c 2(p) = V)pVj + Tr[(I®N ® I A - VjV])p\p0.
REVIEW (6)
The action of the network on a generic M-qubit state |T) is then given by
C2(UfN® Xa )C1(|^ )( xE|) =ufMc2clm m = ufM £r n m m
(7)
the first equality coming from Eq. (5) and the second from the fact that C2 is the inverse of Vy. Clearly, Eq. (7) implies that the fidelity between the output state and the ideal target U fM|'h) is equal to F ^ fJ! independently of g. By Theorem 1, the fidelity is arbitrarily close to one on most input states whenever M grows like 0 {N 2~a). In this case, the number of ancillary qubits used by the protocol scales like M —N + 0(jV'~“/2) [31]. For M growing faster than N 2, the fidelity tends to zero. In summary, given N uses of a completely unknown gate, our network simulates up to N 2 uses with high fidelity on most input states. The simulation works with probability exponentially close to 1, but can fail on some specific inputs: notably, it fails for inputs of the HD form |cp)®M, for which the fidelity with the desired output state is zero. The fact that replication works well in the typical case is analog to other phenomena based on measure concentra tion, such as quantum equilibration [34] and entanglement typicality [35]. State cloning vs state generation.—Deterministic gate replication is not in contradiction with the asymptotic no-cloning theorem. Indeed, suppose that we wanted to use gate replication to clone a completely unknown pure state \y/) = Uv \0) for some fixed state |0). To this purpose, we would have to retrieve the gate Ulf/ from the input state \yf)—a task whose deterministic execution is forbidden by the no-programming theorem [36], Since the state \y/) is arbitrary, a symmetry argument precludes the conversion |i/r) -*■ Uv even probabilistically [19]. Though gate replication cannot be used for state cloning, it may still provide an advantage in the less demanding task of state generation, which consists in producing M copies of the state \yr) from N uses of the gate U,r The advantage does not show up in the universal case, because universal gate replication does not correctly reproduce the action of the gate U®M on the IID input state \0)®M. However, it does show up in nonuniversal cases: for example, Ref. [22] demonstrated that N uses of a phase gate allow one to generate up to N 2 copies of the corresponding phase state. Similarly, we show that universal gate replication allows us to generate M maximally entangled states with fidelity [31 ]
'FgenRV -+ M ]> 1 - 2 (M + 1) exp
N 2' ~2M '
( 8)
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The protocol for generating entangled states also provides an alternative way to generate phase states: given N uses of a phase gate with phase 6 one can first generate M » N approximate copies of the maximally entangled state (|0)|0) + e~,6'|l ) |l ) ) / v /2 and then apply a controlled not gate to each copy and discard the second qubit of the pair, thus, obtaining M » N approximate copies of the state (|0) T e~,e\ 1 ))/s/2. We refer to the ability to produce M » N states from N » 1 uses of the corresponding gate as state supergeneration. Again, we stress that state super generation does not challenge the asymptotic no-cloning theorem, because N uses of a gate cannot be obtained deterministically from N copies of the corresponding state. In the concrete examples of phase states and maximally entangled states, the fidelity of the best deterministic N-to-M cloner scales as (N /M ) 1/2 and as (N /M )3/2, respectively [37], clearly preventing the production of M » N high-fidelity clones. Probabilistic cloning o f maximally entangled states and the optimality o f universal gate replication.—The map U„ —►|4>s) is a one-to-one correspondence between qubit gates and two-qubit maximally entangled states [38,39]. The inverse map -> Ug is implemented by probabi listic teleportation, which succeeds with optimal proba bility 1/4 [40,41]. Combined with the supergeneration of entangled states, the implementability of the map 4>(;) -»■Ug implies that N maximally entangled states can be cloned probabilistically, obtaining up to N 2 high fidelity copies, albeit with exponentially small probability 1/4 N. The result is interesting not only because it provides an explicit protocol achieving superreplication of maximally entangled states, but also because it allows one to prove the optimality of the gate superreplication protocol: if there existed a protocol producing M » N 2 almost perfect copies of a generic unitary gate, such a protocol could be converted into a probabilistic cloning protocol producing M » N 2 almost perfect copies of a generic maximally entangled state. Such a protocol is impossible because it would violate the Heisenberg limit for quantum cloning [19]. Even more strongly, since the fidelity of state replication must vanish for M :» N 2 [19], every gate replication protocol simulating M » N 2 uses must have vanishing entanglement fidelity, and, therefore, vanishing fidelity on most input states. This conclusion applies both to deterministic and probabilistic gate replication protocols. Gate compression.—The argument used to prove gate superreplication can also be applied to the task of gate compression, whose goal is to encode the action of a gate Ux into another gate U f acting on a smaller physical system. Gate compression protocols are useful in distrib uted scenarios wherein a server (Alice) is required to apply the gate Ux to an input state provided by a client (Bob). In such a task, it is natural to minimize the total amount of communication between client and server, by compressing
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of the gate U f N, the protocol allows for an exponential reduction of quantum communication from 2A qubits to 4 log2 A qubits. The amount of quantum communication can be further cut down by nearly a half if a small error is allowed. Indeed, before sending the input to Alice, Bob FIG. 2 (color online). Gate compression. Alice sandwiches the given gate Ux (in green) between two quantum channels A\ and *42 (in red), thus, compressing it into a gate U x acting on a smaller quantum system. The action of U x is retrieved through Bob’s operations B\ and B2 (in blue), which involve the system processed by Alice and a quantum memory kept in Bob’s laboratory. The protocol reduces the amount of quantum com munication needed to simulate the application of Alice’s gate to Bob’s input.
the gate Ux into a gate acting on the smallest possible system. Ideally, the compression should be faithful, in the sense that the action of Ux on the original input state is simulated without errors. In practice, Alice and Bob can often tolerate a small error, especially if this allows them to increase the compression rate. The general form of a gate compression protocol is illustrated in Fig. 2. Here, we consider the scenario where Ux is an A-qubit gate of the IID form U f N, (Jg being an arbitrary rotation of the Bloch sphere. The case of rotations around a fixed axis has been previously considered in [22], Using the decomposition of Eq. (1), the gate U®N can be N/1 put in the block diagonal form U f N — ® (Ug I m n)> ( ■)
7=0
where Ug is a unitary gate acting on the representation space IZj and IMjn is the identity on the multiplicity space A4 jN. The working principle of the compression protocol is to get rid of the multiplicity spaces, on which the gate U f N acts trivially. Specifically, Bob encodes his A'-qubit input into the state of a composite system AB, where system A N/ 2
has Hilbert space H A — ® TZj and system B has Hilbert j
=o
space 7i B = M 0N, the multiplicity space of largest dimen sion. Then, he transmits system A to Alice, keeping system B in his laboratory. On her side, Alice encodes the gate N/
2
,..
U f N into the gate U'g = ® Ug , acting only on system A. i=o
She applies U'g on the input provided by Bob and returns him the output. Finally, Bob applies to systems A and B a joint decoding operation, thus, obtaining an output state of A qubits. All these operations can be devised in such a way that the protocol implements the gate U f N exactly on every /V-qubit input state [31]. The key feature of this protocol is that Alice and Bob only communicate through the exchange of system A, whose dimension grows like A2 instead of 2N. With respect to the naive protocol in which Alice and Bob send to each other the input and the output
can apply the encoding operation E‘f , which compresses the input into a subspace of dimension 0 ( J 2). Setting J = LvWT+I J for some |1) + e~‘Sl\2) + • • • + e~'ed-\ \d - l))/ \/d [31]. Furthermore, N uses of a com pletely unknown gate can be compressed with zero error into a single gate acting on (d - \ ) ( d / 2 + 1) log2 N qubits, a number that can be cut down by nearly a half if one accepts an almost everywhere vanishing error [31]. In conclusion, we showed that A uses of a completely unknown unitary gate allow one to simulate up to A2 uses of the same gate, with high accuracy on all input states except for an exponentially small fraction. The protocol has optimal rate: any attempt to simulate N 2 or more uses is doomed to have vanishing fidelity on most input states. The ability to replicate unitary gates with high fidelity is not in contradiction with the no-cloning theorem, due to the impossibility of deterministically retrieving unitary gates from nonorthogonal pure states. The arguments developed for the gate replication protocol also apply to the task of gate compression, useful in distributed scenarios where a server has to apply a gate to an input state provided by a client. In this scenario, an unknown IID sequence of A unitary gates can be compressed down to a single gate acting only on 0(log2A) qubits, thus, achieving an exponential reduction of quantum communication between client and server. This work is supported by the National Basic Research Program of China Grants No. (973) 2011CBA00300 and No. (2011CBA00301), by the National Natural Science Foundation of China through Grants No. 11450110096, No. 61033001, and No. 61061130540, by the Foundational
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Questions Institute through the Large Grant “The funda mental principles of information dynamics,” and by the 1000 Youth Fellowship Program of China. G. C. is grateful to P. Hayden for pointing out the concentration result for the Schur-Weyl measure in Ref. [26] and to W. Diir for valuable comments.
[1] W. K. Wootters and W. H. Zurek, Nature (London) 299. 802 (1982). [2] D. Dieks, Phys. Lett. 92A, 271 (1982). [3] D. Gottesman, in Quantum Information Science and Its
Contributions to Mathematics: AMS Short Course, Quan tum Computation and Quantum Information, January 3-4, 2009, Washington, DC, Proceedings o f Symposia in Applied Mathematics Vol. 68 (American Mathematical Society, Providence, RI, 2010). [4] D. A. Lidar, Quantum Error Correction (Cambridge University Press, Cambridge, England, 2013). [5] V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, Rev. Mod. Phys. 77, 1225 (2005). [6] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Liitkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009). [7] M. Hillery, V. Buzek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999). [8] R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999). [9] S. Wiesner, SIGACT News 15, 78 (1983). [10] S. Aaronson, in Proceedings o f the 24th Annual IEEE
Conference on Computational Complexity, 2009, CCC ’09, Paris, France (IEEE Computer Society, Los Alamitos, CA, 2009), pp. 229-242. [11] E. Farhi, D. Gosset, A. Hassidim, A. Lutomirski, and P. Shor, in Proceedings o f the 3rd Innovations in Theoreti
cal Computer Science Conference: January 8-10, 2012, Cambridge, MA, USA ITCS ’12 (ACM, New York, 2012), pp. 276-289. [12] A. Molina, T. Vidick, and J. Watrous, in Theory o f quantum computation, communication, and cryptography, edited by K. Iwama,Y. Kawano, and M. Murao, Lecture Notes in Computer Science, Vol. 7582 (Springer, Berlin, 2013), pp. 45-64. [13] V. Buzek and M. Hillery, Phys. Rev. A 54, 1844 (1996). [14] N. Gisin and S. Massar, Phys. Rev. Lett. 79, 2153 (1997). [15] R. F. Werner, Phys. Rev. A 58, 1827 (1998).
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[16] M. Keyl and R. F. Werner, J. Math. Phys. (N.Y.) 40, 3283 (1999). [17] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 80, 4999 (1998). [18] J. Fiurasek, Phys. Rev. A 70, 032308 (2004). [19] G. Chiribella, Y. Yang, and A. C.-C. Yao, Nat. Commun. 4, 2915 (2013). [20] G. Chiribella, G. M. D ’Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 180504 (2008). [21] A. Bisio, G. M. D ’Ariano, P. Perinotti, and M. Sedlak, Phys. Lett. A 378, 1797 (2014). [22] W. Diir, P. Sekatski, and M. Skotiniotis, preceding Letter, Phys. Rev. Lett. 114, 120503 (2015). [23] M. Plesch and V. Buzek, Phys. Rev. A 81, 032317 ( 2010 ).
[24] L. A. Rozema, D. H. Mahler, A. Hayat, P. S. Turner, and A. M. Steinberg, Phys. Rev. Lett. 113, 160504 (2014). [25] D. Bacon, I. L. Chuang, and A. W. Harrow,Phys. Rev. Lett. 97, 170502 (2006). [26] A. W. Harrow, Ph. D. thesis, Massachusetts Institute of Technology, 2005. [27] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 60. 1888 (1999). [28] M. A. Nielsen, Phys. Lett. A 303, 249 (2002). [29] P. Meliot, arXiv: 1009.4034. [30] P. Biane, Int. Math. Res. Not. 2001, 179 (2001). [31] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.ll4.120504 for the technical proofs and for the extension to arbitrary finitedimensional quantum systems, which includes Refs. [32,33], [32] M. Hayashi, Phys. Rev. A 66, 032321 (2002). [33] M. Christandl and G. Mitchison, Commun. Math. Phys. 261, 789 (2006). [34] S. Popescu, A. J. Short, and A. Winter, Nat. Phys. 2, 754 (2006). [35] P. Hayden, D. W. Leung, and A. Winter, Commun. Math. Phys. 265, 95 (2006). [36] M. A. Nielsen and I. L. Chuang, Phys. Rev. Lett. 79, 321 (1997). [37] G. Chiribella and Y. Yang, New J. Phys. 16, 063005 (2014). [38] M.-D. Choi, Linear Algebra Appl. 10, 285 (1975). [39] D. W. Leung, Ph. D. thesis, Stanford University, 2000. [40] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). [41] D. Genkina, G. Chiribella, and L. Hardy, Phys. Rev. A 85, 022330 (2012).
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PRL 114, 120504 (2015)
REVIEW
week ending 27 MARCH 2015
LETTERS
Universal Superreplication of Unitary Gates G. Chiribella, Y. Yang, and C. Huang Center fo r Quantum Information, Institute fo r Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
(Received 3 December 2014; published 25 March 2015) Quantum states obey an asymptotic no-cloning theorem, stating that no deterministic machine can reliably replicate generic sequences of identically prepared pure states. In stark contrast, we show that generic sequences of unitary gates can be replicated deterministically at nearly quadratic rates, with an error vanishing on most inputs except for an exponentially small fraction. The result is not in contradiction with the no-cloning theorem, since the impossibility of deterministically transforming pure states into unitary gates prevents the application of the gate replication protocol to states. In addition to gate replication, we show that N parallel uses of a completely unknown unitary gate can be compressed into a single gate acting on 0(\og2N) qubits, leading to an exponential reduction of the amount of quantum communication needed to implement the gate remotely. DOI: 10.1103/PhysRevLett. 114.120504
PACS numbers: 03.67.Ac, 03.65.Aa
A striking feature of quantum theory is the impossibility of constructing a universal copy machine, which takes, as input, a quantum system in an arbitrary pure state and produces, as output, a number of exact replicas [1,2], Such an impossibility has major implications for quantum error correction [3,4] and cryptographic protocols such as key distribution [5,6], quantum secret sharing [7,8], and quantum money [9-12], The impossibility of universal copy machines is a hard fact: it equally affects deterministic [13-16] and probabi listic machines [17-19], whose performances coincide with those of deterministic machines when it comes to copying completely unknown pure states [18,19], A similar no-go result holds when the universal machine is presented a large number N of identical copies and is required to produce a larger number M > N of approximate replicas: if the replicas have nonvanishing overlap with the desired M - copy state, then the number of extra copies must be negligible compared to N [15], We refer to this fact as the asymptotic no-cloning theorem, expressing the fact that independent and identically distributed (I1D) sequences of pure states cannot be stretched by any significant amount. The asymptotic no-cloning theorem holds also for nonun iversal machines designed to copy continuous sets of states, provided that such machines work deterministically [19]. The impossibility of universal state cloning suggests similar results for quantum gates. Along this line, a no-go theorem for universal gate cloning was proven in Ref. [20], showing that no quantum network can perfectly simulate two uses of an unknown unitary gate by querying it only once. Optimal networks that approximate universal gate cloning were studied in Refs. [20,21], Very recently, Diir and coauthors [22] considered a nonuniversal setup designed to clone phase gates, i.e., gates generated by time evolution with a known Hamiltonian. In this scenario, they devised a quantum network that approximately simulates up to N 2 0031 -9007/15/114( 12)/120504(5)
uses of an unknown phase gate while using it only N times, with vanishing error in the large N limit. Such a result establishes the possibility of superreplication of phase gates— superreplication being the generation of M » N high-fidelity replicas from V » 1 input copies [19], Remarkably, superreplication of phase gates is achieved deterministically, whereas superreplication of phase states has an exponentially small probability of success. The main open question raised by Ref. [22] is whether deterministic superreplication occurs not only for phase gates, but also for arbitrary unitary gates. An affirmative answer would imply that the asymptotic no-cloning theorem only applies to states, whereas it is possible to stretch long IID sequences of reversible gates by up to a quadratic factor. In this Letter, we answer the question in the affirmative, establishing the possibility of universal superreplication of unitary gates, in stark contrast with the asymptotic no-cloning theorem for pure states. Given N uses of a completely unknown unitary gate U, we construct a quan tum network that simulates up to N 2 parallel uses of U, providing an output that is close to the ideal target for all possible input states except for an exponentially small fraction. The quadratic replication rate is optimal: every other network producing replicas at a rate higher than quadratic will necessarily spoil their quality, delivering an output that has vanishing overlap with the output of the desired gate. In addition to replication, we consider the task of gate compression, where the goal is to faithfully encode the action of a black box into a gate operating on a smaller quantum system. For a quantum system of dimension d, we show that N uses of a completely unknown gate can be encoded without any loss into a single gate acting only on (d - l )( d/ 2 + 1)log27V qubits, thus, allowing for an expo nential reduction of computational workspace. The number of qubits can be further cut down by nearly a half if one tolerates an error that vanishes on almost all inputs in the
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© 2015 American Physical Society
PRL 114, 120504 (2015)
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large N limit. The compression of IID gate sequences is the analogue of the compression of IID state sequences [23], recently demonstrated experimentally [24], Universal superreplication of qubit gates. —Let us start from the simple case of qubit gates, represented by unitary matrices in SU(2). A generic gate can be parametrized as Udn = exp{-i0n • j], where 9 G [0, 2k ) is a rotation angle, n = (nx,ny,nz) is a rotation axis, and j = {jxJyJz) is the vector of angular momentum operators (j, —oi/2,i=x,y,z). We define g: — (9. n) and label the unitary gate as Ug. Here, both 9 and n are completely unknown, differing from the setting of [22], where the rotation axis was fixed and only 9 was varying. In order to replicate unknown gates, we consider a network where N parallel uses of Ug are sandwiched between two quantum channels, C\ and C2, as in Fig. 1. The overall action of the network is described by the channel C2(U fN (g) T A)C\, with Ug{-) — Ug - Ufg and I A denoting the identity on a suitable ancillary system. To construct the channels C\ and C2, we decompose the Hilbert space of F qubits (K = N , M) into rotationally invariant subspaces. Choosing F to be even, we have K/2
( 1) 7=0
where j is the quantum number of the total angular momentum, 7Zj is a representation space, of dimension dj = 2j + 1, and M jk is a multiplicity subspace, of dimension mjK. The isomorphism in Eq. (1) is called the Schur transform and can be implemented efficiently in a quantum circuit [25,26]. We now introduce a cutoff on the quantum number j and define the subspace
n f ] = ®{Kj ® M jK).
(2 )
j 0 and to / = N /2 for M growing like N 2 or faster. For channel C2, we choose the inverse of Vj, namely,
where P\K1is the projector on H jK) and p0 is a fixed density matrix with support in '. The key observation is that most F-qubit states are left nearly unchanged by the (K)
channel £) , provided that K is large and J is large compared to \JK. Denoting by F^f'7 ' the fidelity between a generic F-partite pure state T) and its compressed version
S p m m , we have the following. Theorem 1. If |v&) is chosen uniformly at random, then, for every fixed e > 0, the probability that F p J] is smaller than 1 - e satisfies the bound
FIG. 1 (color online). Quantum network for gate replication. The network simulates M parallel uses of an unknown unitary gate Ug, while querying it only N times. The simulation is obtained by transforming the input state of M systems into the joint state of N systems plus an ancilla (via quantum channel C\), applying the unknown gate on the N systems, and then recom bining them with the ancilla via a quantum channel C2, which finally produces M output systems.
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c 2(p) = V)pVj + Tr[(I®N ® I A - VjV])p\p0.
REVIEW (6)
The action of the network on a generic M-qubit state |T) is then given by
C2(UfN® Xa )C1(|^ )( xE|) =ufMc2clm m = ufM £r n m m
(7)
the first equality coming from Eq. (5) and the second from the fact that C2 is the inverse of Vy. Clearly, Eq. (7) implies that the fidelity between the output state and the ideal target U fM|'h) is equal to F ^ fJ! independently of g. By Theorem 1, the fidelity is arbitrarily close to one on most input states whenever M grows like 0 {N 2~a). In this case, the number of ancillary qubits used by the protocol scales like M —N + 0(jV'~“/2) [31]. For M growing faster than N 2, the fidelity tends to zero. In summary, given N uses of a completely unknown gate, our network simulates up to N 2 uses with high fidelity on most input states. The simulation works with probability exponentially close to 1, but can fail on some specific inputs: notably, it fails for inputs of the HD form |cp)®M, for which the fidelity with the desired output state is zero. The fact that replication works well in the typical case is analog to other phenomena based on measure concentra tion, such as quantum equilibration [34] and entanglement typicality [35]. State cloning vs state generation.—Deterministic gate replication is not in contradiction with the asymptotic no-cloning theorem. Indeed, suppose that we wanted to use gate replication to clone a completely unknown pure state \y/) = Uv \0) for some fixed state |0). To this purpose, we would have to retrieve the gate Ulf/ from the input state \yf)—a task whose deterministic execution is forbidden by the no-programming theorem [36], Since the state \y/) is arbitrary, a symmetry argument precludes the conversion |i/r) -*■ Uv even probabilistically [19]. Though gate replication cannot be used for state cloning, it may still provide an advantage in the less demanding task of state generation, which consists in producing M copies of the state \yr) from N uses of the gate U,r The advantage does not show up in the universal case, because universal gate replication does not correctly reproduce the action of the gate U®M on the IID input state \0)®M. However, it does show up in nonuniversal cases: for example, Ref. [22] demonstrated that N uses of a phase gate allow one to generate up to N 2 copies of the corresponding phase state. Similarly, we show that universal gate replication allows us to generate M maximally entangled states with fidelity [31 ]
'FgenRV -+ M ]> 1 - 2 (M + 1) exp
N 2' ~2M '
( 8)
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The protocol for generating entangled states also provides an alternative way to generate phase states: given N uses of a phase gate with phase 6 one can first generate M » N approximate copies of the maximally entangled state (|0)|0) + e~,6'|l ) |l ) ) / v /2 and then apply a controlled not gate to each copy and discard the second qubit of the pair, thus, obtaining M » N approximate copies of the state (|0) T e~,e\ 1 ))/s/2. We refer to the ability to produce M » N states from N » 1 uses of the corresponding gate as state supergeneration. Again, we stress that state super generation does not challenge the asymptotic no-cloning theorem, because N uses of a gate cannot be obtained deterministically from N copies of the corresponding state. In the concrete examples of phase states and maximally entangled states, the fidelity of the best deterministic N-to-M cloner scales as (N /M ) 1/2 and as (N /M )3/2, respectively [37], clearly preventing the production of M » N high-fidelity clones. Probabilistic cloning o f maximally entangled states and the optimality o f universal gate replication.—The map U„ —►|4>s) is a one-to-one correspondence between qubit gates and two-qubit maximally entangled states [38,39]. The inverse map -> Ug is implemented by probabi listic teleportation, which succeeds with optimal proba bility 1/4 [40,41]. Combined with the supergeneration of entangled states, the implementability of the map 4>(;) -»■Ug implies that N maximally entangled states can be cloned probabilistically, obtaining up to N 2 high fidelity copies, albeit with exponentially small probability 1/4 N. The result is interesting not only because it provides an explicit protocol achieving superreplication of maximally entangled states, but also because it allows one to prove the optimality of the gate superreplication protocol: if there existed a protocol producing M » N 2 almost perfect copies of a generic unitary gate, such a protocol could be converted into a probabilistic cloning protocol producing M » N 2 almost perfect copies of a generic maximally entangled state. Such a protocol is impossible because it would violate the Heisenberg limit for quantum cloning [19]. Even more strongly, since the fidelity of state replication must vanish for M :» N 2 [19], every gate replication protocol simulating M » N 2 uses must have vanishing entanglement fidelity, and, therefore, vanishing fidelity on most input states. This conclusion applies both to deterministic and probabilistic gate replication protocols. Gate compression.—The argument used to prove gate superreplication can also be applied to the task of gate compression, whose goal is to encode the action of a gate Ux into another gate U f acting on a smaller physical system. Gate compression protocols are useful in distrib uted scenarios wherein a server (Alice) is required to apply the gate Ux to an input state provided by a client (Bob). In such a task, it is natural to minimize the total amount of communication between client and server, by compressing
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of the gate U f N, the protocol allows for an exponential reduction of quantum communication from 2A qubits to 4 log2 A qubits. The amount of quantum communication can be further cut down by nearly a half if a small error is allowed. Indeed, before sending the input to Alice, Bob FIG. 2 (color online). Gate compression. Alice sandwiches the given gate Ux (in green) between two quantum channels A\ and *42 (in red), thus, compressing it into a gate U x acting on a smaller quantum system. The action of U x is retrieved through Bob’s operations B\ and B2 (in blue), which involve the system processed by Alice and a quantum memory kept in Bob’s laboratory. The protocol reduces the amount of quantum com munication needed to simulate the application of Alice’s gate to Bob’s input.
the gate Ux into a gate acting on the smallest possible system. Ideally, the compression should be faithful, in the sense that the action of Ux on the original input state is simulated without errors. In practice, Alice and Bob can often tolerate a small error, especially if this allows them to increase the compression rate. The general form of a gate compression protocol is illustrated in Fig. 2. Here, we consider the scenario where Ux is an A-qubit gate of the IID form U f N, (Jg being an arbitrary rotation of the Bloch sphere. The case of rotations around a fixed axis has been previously considered in [22], Using the decomposition of Eq. (1), the gate U®N can be N/1 put in the block diagonal form U f N — ® (Ug I m n)> ( ■)
7=0
where Ug is a unitary gate acting on the representation space IZj and IMjn is the identity on the multiplicity space A4 jN. The working principle of the compression protocol is to get rid of the multiplicity spaces, on which the gate U f N acts trivially. Specifically, Bob encodes his A'-qubit input into the state of a composite system AB, where system A N/ 2
has Hilbert space H A — ® TZj and system B has Hilbert j
=o
space 7i B = M 0N, the multiplicity space of largest dimen sion. Then, he transmits system A to Alice, keeping system B in his laboratory. On her side, Alice encodes the gate N/
2
,..
U f N into the gate U'g = ® Ug , acting only on system A. i=o
She applies U'g on the input provided by Bob and returns him the output. Finally, Bob applies to systems A and B a joint decoding operation, thus, obtaining an output state of A qubits. All these operations can be devised in such a way that the protocol implements the gate U f N exactly on every /V-qubit input state [31]. The key feature of this protocol is that Alice and Bob only communicate through the exchange of system A, whose dimension grows like A2 instead of 2N. With respect to the naive protocol in which Alice and Bob send to each other the input and the output
can apply the encoding operation E‘f , which compresses the input into a subspace of dimension 0 ( J 2). Setting J = LvWT+I J for some |1) + e~‘Sl\2) + • • • + e~'ed-\ \d - l))/ \/d [31]. Furthermore, N uses of a com pletely unknown gate can be compressed with zero error into a single gate acting on (d - \ ) ( d / 2 + 1) log2 N qubits, a number that can be cut down by nearly a half if one accepts an almost everywhere vanishing error [31]. In conclusion, we showed that A uses of a completely unknown unitary gate allow one to simulate up to A2 uses of the same gate, with high accuracy on all input states except for an exponentially small fraction. The protocol has optimal rate: any attempt to simulate N 2 or more uses is doomed to have vanishing fidelity on most input states. The ability to replicate unitary gates with high fidelity is not in contradiction with the no-cloning theorem, due to the impossibility of deterministically retrieving unitary gates from nonorthogonal pure states. The arguments developed for the gate replication protocol also apply to the task of gate compression, useful in distributed scenarios where a server has to apply a gate to an input state provided by a client. In this scenario, an unknown IID sequence of A unitary gates can be compressed down to a single gate acting only on 0(log2A) qubits, thus, achieving an exponential reduction of quantum communication between client and server. This work is supported by the National Basic Research Program of China Grants No. (973) 2011CBA00300 and No. (2011CBA00301), by the National Natural Science Foundation of China through Grants No. 11450110096, No. 61033001, and No. 61061130540, by the Foundational
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Questions Institute through the Large Grant “The funda mental principles of information dynamics,” and by the 1000 Youth Fellowship Program of China. G. C. is grateful to P. Hayden for pointing out the concentration result for the Schur-Weyl measure in Ref. [26] and to W. Diir for valuable comments.
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