Quantum Computing and Hidden Variables II: The Complexity of Sampling Histories
ABSTRACT This paper shows that, if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory satisfying a reasonable axiom called "indifference to the identity," we could solve the Graph Isomorphism and Approximate Shortest Vector problems in polynomial time, as well as an oracle problem that is known to require quantum exponential time. We could also search an N-item database using O(N^{1/3}) queries, as opposed to O(N^{1/2}) queries with Grover's search algorithm. On the other hand, the N^{1/3} bound is optimal, meaning that we could probably not solve NP-complete problems in polynomial time. We thus obtain the first good example of a model of computation that appears slightly more powerful than the quantum computing model.
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ABSTRACT: This paper initiates the study of hidden variables from the discrete, abstract perspective of quantum computing. For us, a hidden-variable theory is simply a way to convert a unitary matrix that maps one quantum state to another, into a stochastic matrix that maps the initial probability distribution to the final one in some fixed basis. We list seven axioms that we might want such a theory to satisfy, and then investigate which of the axioms can be satisfied simultaneously. Toward this end, we construct a new hidden-variable theory that is both robust to small perturbations and indi#erent to the identity operation, by exploiting an unexpected connection between unitary matrices and network flows. We also analyze previous hiddenvariable theories of Dieks and Schrodinger in terms of our axioms. In a companion paper, we will show that actually sampling the history of a hidden variable under reasonable axioms is at least as hard as solving the Graph Isomorphism problem; and indeed is probably intractable even for quantum computers.09/2004;
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arXiv:quant-ph/0408119v1 19 Aug 2004
Quantum Computing and Hidden Variables II: The Complexity of
Sampling Histories
Scott Aaronson∗
Abstract
This paper shows that, if we could examine the entire history of a hidden variable, then we
could efficiently solve problems that are believed to be intractable even for quantum computers.
In particular, under any hidden-variable theory satisfying a reasonable axiom called “indiffer-
ence to the identity,” we could solve the Graph Isomorphism and Approximate Shortest Vector
problems in polynomial time, as well as an oracle problem that is known to require quantum
exponential time. We could also search an N-item database using O?N1/3?queries, as opposed
queries with Grover’s search algorithm.
optimal, meaning that we could probably not solve NP-complete problems in polynomial time.
We thus obtain the first good example of a model of computation that appears slightly more
powerful than the quantum computing model.
to O?N1/2?
On the other hand, the N1/3bound is
1Introduction
It is often stressed that hidden-variable theories, such as Bohmian mechanics, yield exactly the
same predictions as ordinary quantum mechanics.
different picture of physical reality, with an additional layer of dynamics beyond that of a state
vector evolving unitarily. This paper addresses a question that, to our knowledge, had never been
raised before: what is the computational complexity of simulating that additional dynamics?
other words, if we could examine a hidden variable’s entire history, then could we solve problems
in polynomial time that are intractable even for quantum computers?
We present strong evidence that the answer is yes.
whether two graphs G and H are isomorphic; while given a basis for a lattice L ∈ Rn, the Approx-
imate Shortest Vector problem asks for a nonzero vector in L within a√n factor of the shortest
one. We show that both problems are efficiently solvable by sampling a hidden variable’s history,
provided the hidden-variable theory satisfies a reasonable axiom that we call “indifference to the
identity operation.” By contrast, despite a decade of effort, neither problem is known to lie in
BQP, the class of problems solvable in quantum polynomial time with bounded error probability.1
Thus, if we let DQP (Dynamical Quantum Polynomial-Time) be the class of problems solvable in
our new model, then this already provides circumstantial evidence that BQP is strictly contained
in DQP.
However, the evidence is stronger than this. For we actually show that DQP contains an entire
class of problems, of which Graph Isomorphism and Approximate Shortest Vector are special
On the other hand, these theories describe a
In
The Graph Isomorphism problem asks
∗University of California, Berkeley. Email: aaronson@cs.berkeley.edu.
1See www.complexityzoo.com for more information about the complexity classes mentioned in this paper.
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cases. Computer scientists know this class as Statistical Zero Knowledge, or SZK.
in previous work [2] we showed that “relative to an oracle,” SZK is not contained in BQP. This is
a technical concept implying that any proof of SZK ⊆ BQP would require techniques unlike those
that are currently known. Combining our result that SZK ⊆ DQP with the oracle separation of [2],
we obtain that BQP ?= DQP relative to an oracle as well. Given computer scientists’ longstanding
inability to separate basic complexity classes, this is nearly the best evidence one could hope for
that sampling histories yields more power than standard quantum computation.
Besides solving SZK problems, we also show that by sampling histories, one could search an
unordered database of N items for a single “marked item” using only O?N1/3?database queries.
algorithms require Θ(N) queries.2On the other hand, we also show that our N1/3upper bound is
the best possible—so even in the histories model, one cannot search an N-item database in (logN)c
steps for some fixed power c.This implies that NP ?⊂ DQP relative to an oracle, which in turn
suggests that DQP is still not powerful enough to solve NP-complete problems in polynomial time.
Note that while Graph Isomorphism and Approximate Shortest Vector are in NP, it is strongly
believed that they are not NP-complete.
At this point we should address a concern that many readers will have.
quantum mechanics by positing the “unphysical” ability to sample histories, isn’t it completely
unsurprising if we can then solve problems that were previously intractable?
answer is no, for three reasons.
First, almost every change that makes the quantum computing model more powerful, seems
to make it so much more powerful that NP-complete and even harder problems become solvable
efficiently. To give some examples, NP-complete problems can be solved in polynomial time using
a nonlinear Schr¨ odinger equation, as shown by Abrams and Lloyd [4]; using closed timelike curves,
as shown by Bacon [6]; or using a measurement rule of the form |ψ|pfor any p ?= 2, as shown by
us [3]. It is also easy to see that we could solve NP-complete problems if, given a quantum state
|ψ?, we could request a classical description of |ψ?, such as a list of amplitudes or a preparation
procedure.3
By contrast, ours is the first independently motivated model we know of that seems
more powerful than quantum computing, but only slightly so.4
unordered search in our model takes about N1/3steps, as compared to N steps classically and
N1/2quantum-mechanically, suggests that DQP somehow “continues a sequence” that begins with
P and BQP. It would be interesting to find a model in which search takes N1/4or N1/5steps.
The second reason our results are surprising is that, given a hidden variable, the distribution
over its possible values at any single time is governed by standard quantum mechanics, and is
therefore efficiently samplable on a quantum computer.
confers any extra computational power, then it can only be because of correlations between the
variable’s values at different times.
The third reason is our criterion for success.
Graph Isomorphism under some hidden-variable theory; or even that, under any theory satisfying
Furthermore,
By comparison, Grover’s quantum search algorithm [11] requires Θ?N1/2?queries, while classical
Once we extend
We believe the
Moreover, the striking fact that
So if examining the variable’s history
We are not saying merely that one can solve
2For readers unfamiliar with asymptotic notation: O(f (N)) means “at most order f (N),” Ω(f (N)) means “at
least order f (N),” and Θ(f (N)) means “exactly order f (N).”
3For as Abrams and Lloyd [4] observed, we can so arrange things that |ψ? = |0? if an NP-complete instance of
interest to us has no solution, but |ψ? =√1 − ε|0? +√ε|1? for some tiny ε if it has a solution.
4One can define other, less motivated, models with the same property by allowing “non-collapsing measurements”
of quantum states, but these models are very closely related to ours. Indeed, a key ingredient of our results will be
to show that certain kinds of non-collapsing measurements can be simulated using histories.
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the indifference axiom, there exists an algorithm to solve it; but rather that there exists a single
algorithm that solves Graph Isomorphism under any theory satisfying indifference. Thus, we must
consider even theories that are specifically designed to thwart such an algorithm.
But what is the motivation for our results? The first motivation is that, within the community
of physicists who study hidden-variable theories such as Bohmian mechanics, there is great interest
in actually calculating the hidden-variable trajectories for specific physical systems [15, 12]. Our
results show that, when many interacting particles are involved, this task might be fundamentally
intractable, even if a quantum computer is available.
computer science, studying “unrealistic” models of computation has often led to new insights into
realistic ones; and likewise we expect that the DQP model could lead to new results about standard
quantum computation. Indeed, in a sense this has already happened. For our result that SZK ?⊂
BQP relative to an oracle [2] grew out of work on the BQP versus DQP question. Yet the “quantum
lower bound for the collision problem” underlying that result provided the first evidence that
cryptographic hash functions could be secure against quantum attack, and ruled out a large class
of possible quantum algorithms for Graph Isomorphism, Approximate Shortest Vector, and related
problems.
The second motivation is that, in classical
1.1 Outline of Paper
The precise definition of a hidden-variable theory that we use in this paper was developed in a
companion paper [1]. Familiarity with [1] is helpful but not essential for understanding this paper.
In Section 2, we review the relevant concepts from [1], and then formally define DQP as the class
of problems solvable by a classical polynomial-time algorithm with access to a “history oracle.”
Given a sequence of quantum circuits as input, this oracle returns a sample from a corresponding
distribution over histories of a hidden variable, according to some hidden-variable theory T . The
oracle can choose T “adversarially,” subject to two constraints: T must be robust to small errors
(since otherwise the definition of DQP could depend on the choice of gate set), and it must satisfy
the indifference axiom.
So what is the indifference axiom, then?
|ψ? ∈ HA⊗ HB (entangled or unentangled), if a unitary operation acts only on the HApart of
|ψ? (i.e. has the form U ⊗ I), then the hidden-variable transitions can also only involve the HA
part. Note that this is quite different from locality in the sense of Bell’s theorem: the probability of
transitioning between two basis states |xA?⊗|xB? and |yA?⊗|xB? can depend on the complete state
|ψ?; all we require is that if xB ?= yB, then the probability of transitioning between |xA? ⊗ |xB?
and |yA? ⊗ |yB? is zero.
Bohmian mechanics. However, to us it simply expresses the idea that, if we have a state such as
(|a? + |b? + |c? + |d?)/2, and a partial measurement yields a new state
|a? + |b?
2
Intuitively it says that, given a bipartite state
Indifference is a substantive axiom, and is violated (for example) by
|Rab? +|c? + |d?
2
|Rcd?,
where |Rab? and |Rcd? denote two configurations of a recording apparatus, then so long as we leave
the recording apparatus alone, all further hidden-variable transitions should be between |a? and
|b? or between |c? and |d?, not between (say) |a? and |c?.
would need some other way to rule out the degenerate hidden-variable theory, which takes the
hidden-variable values at different times to be completely independent of one another. Were this
“product theory” allowed, we would have DQP = BQP for trivial reasons.
If we abandoned this axiom, then we
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An earlier version of this paper required another axiom—symmetry under permutations of basis
states—which seems much harder to justify than indifference.
to eliminate the dependence of our algorithms on the symmetry axiom.
Section 3 establishes the most basic facts about DQP: for example, that BQP ⊆ DQP, and that
DQP is independent of the choice of gate set. Then Section 4 presents the “juggle subroutine,” a
crucial ingredient in both main algorithms of the paper. Given a state of the form (|a? + |b?)/√2
or (|a? − |b?)/√2, the goal of this subroutine is to “juggle” a hidden variable between |a? and |b?,
so that when we inspect the hidden variable’s history, both |a? and |b? are observed with high
probability. The difficulty is that this needs to work under any indifferent hidden-variable theory.
Next, Section 5 combines the juggle subroutine with a technique of Valiant and Vazirani [19]
to prove that SZK ⊆ DQP, from which it follows in particular that Graph Isomorphism and Ap-
proximate Shortest Vector are in DQP. Then Section 6 applies the juggle subroutine to search
an N-item database in O?N1/3?queries, and also proves that this N1/3bound is optimal.
conclude in Section 7 with some directions for further research.
However, we have since been able
We
2 The Computational Model
We now explain our model of computation, building our way up to the complexity class DQP. Our
starting point is the definition of hidden-variable theory that we gave in [1].
paper: for us a hidden-variable theory is simply a family of functions {SN}N∈{1,2,...}, where each
SN maps an N × N density matrix ρ and an N × N unitary matrix U onto an N × N stochastic
matrix S = SN(ρ,U). In this paper, ρ will always be a pure state of l = log2N qubits. That is,
ρ = |ψ??ψ| where
|ψ? =
x∈{0,1}l
To recap from that
?
αx|x?.
What is essential is that S map the probability distribution induced by measuring |ψ? in the
computational basis {|x?}x∈{0,1}l, onto the probability distribution induced by measuring U |ψ? in
that same basis.More formally, let (M)xydenote the entry in the xthcolumn and ythrow of
matrix M, and let
U |ψ? =
x∈{0,1}l
?
βx|x?.
Then we require that for all y ∈ {0,1}l,
?
x∈{0,1}l
(S)xy|αx|2= |βy|2.
It is clear that there are infinitely many theories satisfying the above marginalization axiom; the
simplest one is the product theory PT , which sets (S)xy= |βy|2for all x,y. To narrow down the
choices, in [1] we proposed seven additional axioms that we might want any hidden-variable theory
to satisfy. We then showed that, although not all of the axioms can be satisfied simultaneously, two
of the most important ones—called indifference and robustness—can be satisfied simultaneously.
Let us restate those two axioms in the present context. Indifference says that if U is generalized
block-diagonal (i.e. a permutation of a block-diagonal matrix), then S is also generalized block-
diagonal with the same block structure or some refinement thereof. So in particular, if |ψ? belongs
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to a tensor-product Hilbert space HA⊗HB, and if U acts only on HA(i.e. never maps a basis state
|xA? ⊗ |xB? to |yA? ⊗ |yB? where xB?= yB), then S (|ψ?,U) acts only on HAas well. Robustness
says that S is insensitive to small perturbations of |ψ? or U.
we call a theory robust if for all b > 0, there exists c > 0 such that for all l, all pairs of states
|ψ? =?
such that
xy
????(S)xy|αx|2−
for all x,y, where S = S (|ψ?,U) and?S = S
It is easy to show that the product theory PT satisfies robustness but not indifference. In [1],
we analyzed three other hidden-variable theories: the Dieks theory DT , which satisfies indifference
but not robustness; the flow theory FT , which satisfies both indifference and robustness; and the
Schr¨ odinger theory ST , which satisfies indifference, and which we conjecture satisfies robustness.
The details of those theories are mostly irrelevant for this paper. Indeed, our algorithms will work
under any hidden-variable theory that satisfies the indifference axiom.
take into account that even in theory (let alone in practice), a generic unitary cannot be represented
exactly with a finite universal gate set, only approximated arbitrarily well, then we also need the
robustness axiom. Thus, a key result from [1] that we rely on is that there exists a hidden-variable
theory (namely FT ) satisfying both indifference and robustness.
Let a quantum computer have the initial state |0?⊗l, and suppose we apply a sequence U =
(U1,...,UT) of unitary operations, each of which is implemented by a polynomial-size quantum
circuit. Then a history of a hidden variable through the computation is a sequence H = (v0,...,vT)
of basis states, where vtis the variable’s value immediately after Utis applied (thus v0= |0?⊗l).
Given any hidden-variable theory T , we can obtain a probability distribution Ω(U,T ) over histories
by just applying T repeatedly, once for each Ut, to obtain the stochastic matrices
?
Note that Ω(U,T ) is a Markov distribution; that is, each vt is independent of the other vi’s
conditioned on vt−1and vt+1. Admittedly, Ω(U,T ) could depend on the precise way in which the
combined circuit UT···U1is “sliced” into component circuits U1,...,UT. But as we showed in [1],
such dependence on the granularity of unitaries is unavoidable in any hidden-variable theory other
than PT .
Given a hidden-variable theory T , let O(T ) be an oracle that takes as input a positive integer
l, and a sequence of quantum circuits U = (U1,...,UT) that act on l qubits.
specified by a sequence?gt,1,...,gt,m(t)
oracle O(T ) returns as output a sample (v0,...,vT) from the history distribution Ω(U,T ) defined
previously. Now let A be a deterministic classical Turing machine that is given oracle access to
O(T ). The machine A receives an input x, makes a single oracle query to O(T ), then produces
an output based on the response.We say a set of strings L is in DQP if there exists an A such
that for all sufficiently large n and inputs x ∈ {0,1}n, and all theories T satisfying the indifference
and robustness axioms, A correctly decides whether x ∈ L with probability at least 2/3, in time
polynomial in n.
To make this intuition formal,
x∈{0,1}l αx|x? and
????(U)xy−
????ψ
?
=?
x∈{0,1}l ? αx|x? such that
?
ψ|?ψ
?
≥ 1 − 2−cl, and all U and?U
??U
?
????≤ 2−clfor all x,y, we have
??S
?
xy|? αx|2
,?U
????≤ 2−bl
?????ψ
??
.
On the other hand, if we
S|0?⊗l,U1
?
, S
?
U1|0?⊗l,U2
?
, ... S
?
UT−1···U1|0?⊗l,UT
?
.
Here each Ut is
?of gates chosen from some finite universal gate set G. The
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